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1 Polnomial Functions.1 Graphing Polnomial Functions. Adding, Subtracting, and Multipling Polnomials.3 Dividing Polnomials. Factoring Polnomials.5 Solving Polnomial Equations. The Fundamental Theorem of Algebra.7 Transformations of Polnomial Functions.8 Analzing Graphs of Polnomial Functions.9 Modeling with Polnomial Functions SEE the Big Idea Quonset Hut (p. 18) Zebra Mussels (p. 03) Ruins of Caesarea (p. 195) Electric Vehicles (p. 11) Basketball (p. 178)

2 Maintaining Mathematical Proficienc Simplifing Algebraic Epressions Eample 1 Simplif the epression = (9 + ) Distributive Propert = 13 Add coefficients. Eample Simplif the epression ( + ) + 3( ). ( + ) + 3( ) = () + () + 3() + 3( ) Distributive Propert = Multipl. = Group like terms. = + Combine like terms. Simplif the epression m m 7m ( + ). 9 ( 1) 5. (z + ) (1 z) Finding Volume Eample 3 Find the volume of a rectangular prism with length 10 centimeters, width centimeters, and height 5 centimeters. Volume = wh Write the volume formula. 10 cm 5 cm cm = (10)()(5) Substitute 10 for, for w, and 5 for h. = 00 Multipl. The volume is 00 cubic centimeters. Find the volume of the solid. 7. cube with side length inches 8. sphere with radius feet 9. rectangular prism with length feet, width feet, and height feet 10. right clinder with radius 3 centimeters and height 5 centimeters 11. ABSTRACT REASONING Does doubling the volume of a cube have the same effect on the side length? Eplain our reasoning. Dnamic Solutions available at BigIdeasMath.com 155

3 Mathematical Practices Mathematicall profi cient students use technological tools to eplore concepts. Using Technolog to Eplore Concepts Core Concept Continuous Functions A function is continuous when its graph has no breaks, holes, or gaps. Graph of a continuous function Graph of a function that is not continuous Determining Whether Functions Are Continuous Use a graphing calculator to compare the two functions. What can ou conclude? Which function is not continuous? f() = g() = 3 1 The graphs appear to be identical, but g is not defined when = 1. There is a hole in the graph of g at the point (1, 1). Using the table feature of a graphing calculator, ou obtain an error for g() when = 1. So, g is not continuous. 3 f() = 3 3 hole 3 g() = 3 1 Monitoring Progress Use a graphing calculator to determine whether the function is continuous. Eplain our reasoning. 1. f() = X Y1=1. f() = + 5. f() = 1. f() = f() = f() = 1 7. f() = 8. f() = 3 9. f() = Y X Y ERROR Y1=ERROR 15 Chapter Polnomial Functions

4 .1 Graphing Polnomial Functions Essential Question What are some common characteristics of the graphs of cubic and quartic polnomial functions? A polnomial function of the form f() = a n n + a n 1 n a 1 + a 0 where a n 0, is cubic when n = 3 and quartic when n =. Identifing Graphs of Polnomial Functions Work with a partner. Match each polnomial function with its graph. Eplain our reasoning. Use a graphing calculator to verif our answers. a. f() = 3 b. f() = 3 + c. f() = + 1 d. f() = e. f() = 3 f. f() = A. B. C. D. E. F. Identifing -Intercepts of Polnomial Graphs CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to justif our conclusions and communicate them to others. Work with a partner. Each of the polnomial graphs in Eploration 1 has -intercept(s) of 1, 0, or 1. Identif the -intercept(s) of each graph. Eplain how ou can verif our answers. Communicate Your Answer 3. What are some common characteristics of the graphs of cubic and quartic polnomial functions?. Determine whether each statement is true or false. Justif our answer. a. When the graph of a cubic polnomial function rises to the left, it falls to the right. b. When the graph of a quartic polnomial function falls to the left, it rises to the right. Section.1 Graphing Polnomial Functions 157

5 .1 Lesson Core Vocabular polnomial, p. 158 polnomial function, p. 158 end behavior, p. 159 Previous monomial linear function quadratic function What You Will Learn Identif polnomial functions. Graph polnomial functions using tables and end behavior. Polnomial Functions Recall that a monomial is a number, a variable, or the product of a number and one or more variables with whole number eponents. A polnomial is a monomial or a sum of monomials. A polnomial function is a function of the form f() = a n n + a n 1 n a 1 + a 0 where a n 0, the eponents are all whole numbers, and the coefficients are all real numbers. For this function, a n is the leading coefficient, n is the degree, and a 0 is the constant term. A polnomial function is in standard form when its terms are written in descending order of eponents from left to right. You are alread familiar with some tpes of polnomial functions, such as linear and quadratic. Here is a summar of common tpes of polnomial functions. Common Polnomial Functions Degree Tpe Standard Form Eample 0 Constant f() = a 0 f() = 1 1 Linear f() = a 1 + a 0 f() = 5 7 Quadratic f() = a + a 1 + a 0 f() = Cubic f() = a a + a 1 + a 0 f() = Quartic f() = a + a a + a 1 + a 0 f() = + 1 Identifing Polnomial Functions Decide whether each function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient. a. f() = b. g() = c. h() = d. k() = + 3 a. The function is a polnomial function that is alread written in standard form. It has degree 3 (cubic) and a leading coefficient of. b. The function is a polnomial function written as g() = in standard form. It has degree (quartic) and a leading coefficient of. c. The function is not a polnomial function because the term 7 1 has an eponent that is not a whole number. d. The function is not a polnomial function because the term 3 does not have a variable base and an eponent that is a whole number. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Decide whether the function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient. 1. f() = p() = q() = Chapter Polnomial Functions

6 Evaluating a Polnomial Function Evaluate f() = when = 3. f() = Write original equation. f(3) = (3) 8(3) + 5(3) 7 Substitute 3 for. = Evaluate powers and multipl. = 98 Simplif. The end behavior of a function s graph is the behavior of the graph as approaches positive infinit (+ ) or negative infinit ( ). For the graph of a polnomial function, the end behavior is determined b the function s degree and the sign of its leading coefficient. READING The epression + is read as approaches positive infinit. Core Concept End Behavior of Polnomial Functions Degree: odd Degree: odd Leading coefficient: positive Leading coefficient: negative f() as f() + as + f() + as f() as + Degree: even Leading coefficient: positive Degree: even Leading coefficient: negative f() + as f() + as + f() as f() as + Describing End Behavior Check 10 Describe the end behavior of the graph of f() = The function has degree and leading coefficient 0.5. Because the degree is even and the leading coefficient is negative, f() as and f() as +. Check this b graphing the function on a graphing calculator, as shown. 10 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Evaluate the function for the given value of.. f() = ; = 5. f() = ; =. Describe the end behavior of the graph of f() = Section.1 Graphing Polnomial Functions 159

7 Graphing Polnomial Functions To graph a polnomial function, first plot points to determine the shape of the graph s middle portion. Then connect the points with a smooth continuous curve and use what ou know about end behavior to sketch the graph. Graphing Polnomial Functions Graph (a) f() = and (b) f() = 3 +. a. To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior f() The degree is odd and the leading coefficient is negative. So, f() + as and f() as +. b. To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior f() 1 0 The degree is even and the leading coefficient is positive. So, f() + as and f() + as +. (, 3) 3 1 (1, 0) (, 1) ( 1, ) (0, 3) (0, ) ( 1, ) 1 (1, 0) (, ) Sketching a Graph Sketch a graph of the polnomial function f having these characteristics. f is increasing when < 0 and >. f is decreasing when 0 < <. f() > 0 when < < 3 and > 5. f() < 0 when < and 3 < < 5. Use the graph to describe the degree and leading coefficient of f. increasing decreasing increasing The graph is above the -ais when f() > The graph is below the -ais when f() < Chapter Polnomial Functions From the graph, f() as and f() + as +. So, the degree is odd and the leading coefficient is positive.

8 Solving a Real-Life Problem The estimated number V (in thousands) of electric vehicles in use in the United States can be modeled b the polnomial function V(t) = t t t.01 where t represents the ear, with t = 1 corresponding to 001. a. Use a graphing calculator to graph the function for the interval 1 t 10. Describe the behavior of the graph on this interval. b. What was the average rate of change in the number of electric vehicles in use from 001 to 010? c. Do ou think this model can be used for ears before 001 or after 010? Eplain our reasoning. a. Using a graphing calculator and a viewing 5 window of 1 10 and 0 5, ou obtain the graph shown. From 001 to 00, the numbers of electric vehicles in use increased. Around 005, the growth in the numbers in use slowed and 1 started to level off. Then the numbers in use 0 started to increase again in 009 and 010. b. The ears 001 and 010 correspond to t = 1 and t = 10. Average rate of change over 1 t 10: V(10) V(1) =.3 9 The average rate of change from 001 to 010 is about. thousand electric vehicles per ear. c. Because the degree is odd and the leading coefficient is positive, V(t) as t and V(t) + as t +. The end behavior indicates that the model has unlimited growth as t increases. While the model ma be valid for a few ears after 010, in the long run, unlimited growth is not reasonable. Notice in 000 that V(0) =.01. Because negative values of V(t) do not make sense given the contet (electric vehicles in use), the model should not be used for ears before Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the polnomial function. 7. f() = f() = 3 9. f() = Sketch a graph of the polnomial function f having these characteristics. f is decreasing when < 1.5 and >.5; f is increasing when 1.5 < <.5. f() > 0 when < 3 and 1 < < ; f() < 0 when 3 < < 1 and >. Use the graph to describe the degree and leading coefficient of f. 11. WHAT IF? Repeat Eample using the alternative model for electric vehicles of V(t) = t t t t Section.1 Graphing Polnomial Functions 11

9 .1 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Eplain what is meant b the end behavior of a polnomial function.. WHICH ONE DOESN T BELONG? Which function does not belong with the other three? Eplain our reasoning. f() = g() = h() = k() = Monitoring Progress and Modeling with Mathematics In Eercises 3 8, decide whether the function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient. (See Eample 1.) 3. f() = p() = f() = g() = h() = h() = ERROR ANALYSIS In Eercises 9 and 10, describe and correct the error in analzing the function. 9. f() = f is a polnomial function. The degree is 3 and f is a cubic function. The leading coefficient is 8. In Eercises 11 1, evaluate the function for the given value of. (See Eample.) 11. h() = ; = 1. f() = ; = g() = ; = 8 1. g() = ; = p() = ; = 1 1. h() = ; = In Eercises 17 0, describe the end behavior of the graph of the function. (See Eample 3.) 17. h() = g() = f() = f() = In Eercises 1 and, describe the degree and leading coefficient of the polnomial function using the graph f() = f is a polnomial function. The degree is and f is a quartic function. The leading coefficient is. 1 Chapter Polnomial Functions

10 3. USING STRUCTURE Determine whether the function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient. f() = WRITING Let f() = 13. State the degree, tpe, and leading coefficient. Describe the end behavior of the function. Eplain our reasoning. In Eercises 5 3, graph the polnomial function. (See Eample.) 5. p() = 3. g() = f() = p() = h() = h() = g() = p() = ANALYZING RELATIONSHIPS In Eercises 33 3, describe the -values for which (a) f is increasing or decreasing, (b) f() > 0, and (c) f() < f 3. 8 f 38. f is increasing when < < 3; f is decreasing when < and > 3. f() > 0 when < and 1 < < 5; f() < 0 when < < 1 and > f is increasing when < < 0 and > ; f is decreasing when < and 0 < <. f() > 0 when < 3, 1 < < 1, and > 3; f() < 0 when 3 < < 1 and 1 < < f is increasing when < 1 and > 1; f is decreasing when 1 < < 1. f() > 0 when 1.5 < < 0 and > 1.5; f() < 0 when < 1.5 and 0 < < MODELING WITH MATHEMATICS From 1980 to 007 the number of drive-in theaters in the United States can be modeled b the function d(t) = 0.11t t 3.5t + 1 where d(t) is the number of open theaters and t is the number of ears after (See Eample.) a. Use a graphing calculator to graph the function for the interval 0 t 7. Describe the behavior of the graph on this interval. b. What is the average rate of change in the number of drive-in movie theaters from 1980 to 1995 and from 1995 to 007? Interpret the average rates of change. c. Do ou think this model can be used for ears before 1980 or after 007? Eplain f 3. f 1 In Eercises 37 0, sketch a graph of the polnomial function f having the given characteristics. Use the graph to describe the degree and leading coefficient of the function f. (See Eample 5.) 37. f is increasing when > 0.5; f is decreasing when < 0.5. f() > 0 when < and > 3; f() < 0 when < < 3.. PROBLEM SOLVING The weight of an ideal round-cut diamond can be modeled b w = d d + 0.0d 0.01 where w is the weight of the diameter diamond (in carats) and d is the diameter (in millimeters). According to the model, what is the weight of a diamond with a diameter of 1 millimeters? Section.1 Graphing Polnomial Functions 13

11 3. ABSTRACT REASONING Suppose f() as and f() as. Describe the end behavior of g() = f(). Justif our answer.. THOUGHT PROVOKING Write an even degree polnomial function such that the end behavior of f is given b f() as and f() as. Justif our answer b drawing the graph of our function. 5. USING TOOLS When using a graphing calculator to graph a polnomial function, eplain how ou know when the viewing window is appropriate.. MAKING AN ARGUMENT Your friend uses the table to speculate that the function f is an even degree polnomial and the function g is an odd degree polnomial. Is our friend correct? Eplain our reasoning. f() g() DRAWING CONCLUSIONS The graph of a function is smmetric with respect to the -ais if for each point (a, b) on the graph, ( a, b) is also a point on the graph. The graph of a function is smmetric with respect to the origin if for each point (a, b) on the graph, ( a, b) is also a point on the graph. a. Use a graphing calculator to graph the function = n when n = 1,, 3,, 5, and. In each case, identif the smmetr of the graph. b. Predict what smmetr the graphs of = 10 and = 11 each have. Eplain our reasoning and then confirm our predictions b graphing. 8. HOW DO YOU SEE IT? The graph of a polnomial function is shown. f a. Describe the degree and leading coefficient of f. b. Describe the intervals where the function is increasing and decreasing. c. What is the constant term of the polnomial function? 9. REASONING A cubic polnomial function f has a leading coefficient of and a constant term of 5. When f(1) = 0 and f() = 3, what is f( 5)? Eplain our reasoning. 50. CRITICAL THINKING The weight (in pounds) of a rainbow trout can be modeled b = , where is the length (in inches) of the trout. a. Write a function that relates the weight and length of a rainbow trout when is measured in kilograms and is measured in centimeters. Use the fact that 1 kilogram.0 pounds and 1 centimeter 0.39 inch. b. Graph the original function and the function from part (a) in the same coordinate plane. What tpe of transformation can ou appl to the graph of = to produce the graph from part (a)? Maintaining Mathematical Proficienc Simplif the epression. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons h 3 g + 3hg 3 + 7h g + 5h 3 g + hg wk + 3kz kw + 9zk kw 5. a (m 7a 3 ) m(a 10) 55. 3( ) + 3( + 3) ( 1) 5. cv(9 3c) + c(v c) + c 1 Chapter Polnomial Functions

12 . Adding, Subtracting, and Multipling Polnomials Essential Question How can ou cube a binomial? Cubing Binomials Work with a partner. Find each product. Show our steps. a. ( + 1) 3 = ( + 1)( + 1) Rewrite as a product of first and second powers. = ( + 1) Multipl second power. = Multipl binomial and trinomial. = Write in standard form, a 3 + b + c + d. b. (a + b) 3 = (a + b)(a + b) Rewrite as a product of first and second powers. = (a + b) Multipl second power. = Multipl binomial and trinomial. = Write in standard form. c. ( 1) 3 = ( 1)( 1) Rewrite as a product of first and second powers. = ( 1) Multipl second power. = Multipl binomial and trinomial. = Write in standard form. d. (a b) 3 = (a b)(a b) Rewrite as a product of first and second powers. = (a b) Multipl second power. LOOKING FOR STRUCTURE To be proficient in math, ou need to look closel to discern a pattern or structure. = Multipl binomial and trinomial. = Write in standard form. Generalizing Patterns for Cubing a Binomial Work with a partner. 1 a. Use the results of Eploration 1 to describe a pattern for the coefficients of the terms when ou epand the cube of a 1 1 binomial. How is our pattern related to Pascal s Triangle, 1 1 shown at the right? b. Use the results of Eploration 1 to describe a pattern for the eponents of the terms in the epansion of a cube of a binomial. 1 1 c. Eplain how ou can use the patterns ou described in parts (a) and (b) to find the product ( 3) 3. Then find this product. Communicate Your Answer 3. How can ou cube a binomial?. Find each product. a. ( + ) 3 b. ( ) 3 c. ( 3) 3 d. ( 3) 3 e. ( + 3) 3 f. (3 5) 3 Section. Adding, Subtracting, and Multipling Polnomials 15

13 . Lesson What You Will Learn Core Vocabular Pascal s Triangle, p. 19 Previous like terms identit Add and subtract polnomials. Multipl polnomials. Use Pascal s Triangle to epand binomials. Adding and Subtracting Polnomials Recall that the set of integers is closed under addition and subtraction because ever sum or difference results in an integer. To add or subtract polnomials, ou add or subtract the coefficients of like terms. Because adding or subtracting polnomials results in a polnomial, the set of polnomials is closed under addition and subtraction. Adding Polnomials Verticall and Horizontall a. Add and in a vertical format. b. Add and 5 + in a horizontal format. a b. ( ) + ( 5 + ) = = To subtract one polnomial from another, add the opposite. To do this, change the sign of each term of the subtracted polnomial and then add the resulting like terms. COMMON ERROR A common mistake is to forget to change signs correctl when subtracting one polnomial from another. Be sure to add the opposite of ever term of the subtracted polnomial. Subtracting Polnomials Verticall and Horizontall a. Subtract from in a vertical format. b. Subtract 3z + z from z + 3z in a horizontal format. a. Align like terms, then add the opposite of the subtracted polnomial ( ) b. Write the opposite of the subtracted polnomial, then add like terms. (z + 3z) (3z + z ) = z + 3z 3z z + = z + z + Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the sum or difference. 1. ( + 5) + (7 9). (3t 3 + 8t t ) (5t 3 t + 17) 1 Chapter Polnomial Functions

14 Multipling Polnomials To multipl two polnomials, ou multipl each term of the first polnomial b each term of the second polnomial. As with addition and subtraction, the set of polnomials is closed under multiplication. Multipling Polnomials Verticall and Horizontall REMEMBER Product of Powers Propert a m a n = a m+n a is a real number and m and n are integers. a. Multipl + + and 3 in a vertical format. b. Multipl + 5 and 3 + in a horizontal format. a Multipl + + b Multipl + + b Combine like terms. b. ( + 5)(3 + ) = ( + 5)3 ( + 5) + ( + 5) = = Multipling Three Binomials Multipl 1, +, and + 5 in a horizontal format. ( 1)( + )( + 5) = ( + 3 )( + 5) = ( + 3 ) + ( + 3 )5 = = Some binomial products occur so frequentl that it is worth memorizing their patterns. You can verif these polnomial identities b multipling. COMMON ERROR In general, (a ± b) a ± b and (a ± b) 3 a 3 ± b 3. Core Concept Special Product Patterns Sum and Difference Eample (a + b)(a b) = a b ( + 3)( 3) = 9 Square of a Binomial Eample (a + b) = a + ab + b ( + ) = (a b) = a ab + b (t 5) = t 0t + 5 Cube of a Binomial Eample (a + b) 3 = a 3 + 3a b + 3ab + b 3 (z + 3) 3 = z 3 + 9z + 7z + 7 (a b) 3 = a 3 3a b + 3ab b 3 (m ) 3 = m 3 m + 1m 8 Section. Adding, Subtracting, and Multipling Polnomials 17

15 Proving a Polnomial Identit a. Prove the polnomial identit for the cube of a binomial representing a sum: (a + b) 3 = a 3 + 3a b + 3ab + b 3. b. Use the cube of a binomial in part (a) to calculate a. Epand and simplif the epression on the left side of the equation. (a + b) 3 = (a + b)(a + b)(a + b) = (a + ab + b )(a + b) = (a + ab + b )a + (a + ab + b )b = a 3 + a b + a b + ab + ab + b 3 = a 3 + 3a b + 3ab + b 3 The simplified left side equals the right side of the original identit. So, the identit (a + b) 3 = a 3 + 3a b + 3ab + b 3 is true. b. To calculate 11 3 using the cube of a binomial, note that 11 = = (10 + 1) 3 Write 11 as = (10) (1) + 3(10)(1) Cube of a binomial = Epand. = 1331 Simplif. REMEMBER Power of a Product Propert (ab) m = a m b m a and b are real numbers and m is an integer. Using Special Product Patterns Find each product. a. (n + 5)(n 5) b. (9 ) c. (ab + ) 3 a. (n + 5)(n 5) = (n) 5 Sum and difference = 1n 5 Simplif. b. (9 ) = (9) (9)() + Square of a binomial = Simplif. c. (ab + ) 3 = (ab) 3 + 3(ab) () + 3(ab)() + 3 Cube of a binomial = a 3 b 3 + 1a b + 8ab + Simplif. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the product. 3. ( + 5)( + 1). ( )( ) 5. (m )(m 1)(m + 3). (3t )(3t + ) 7. (5a + ) 8. ( 3) 3 9. (a) Prove the polnomial identit for the cube of a binomial representing a difference: (a b) 3 = a 3 3a b + 3ab b 3. (b) Use the cube of a binomial in part (a) to calculate Chapter Polnomial Functions

16 Core Concept Pascal s Triangle Pascal s Triangle Consider the epansion of the binomial (a + b) n for whole number values of n. When ou arrange the coefficients of the variables in the epansion of (a + b) n, ou will see a special pattern called Pascal s Triangle. Pascal s Triangle is named after French mathematician Blaise Pascal (13 1). In Pascal s Triangle, the first and last numbers in each row are 1. Ever number other than 1 is the sum of the closest two numbers in the row directl above it. The numbers in Pascal s Triangle are the same numbers that are the coefficients of binomial epansions, as shown in the first si rows. n (a + b) n Binomial Epansion Pascal s Triangle 0th row 0 (a + b) 0 = 1 1 1st row 1 (a + b) 1 = 1a + 1b 1 1 nd row (a + b) = 1a + ab + 1b 1 1 3rd row 3 (a + b) 3 = 1a 3 + 3a b + 3ab + 1b th row (a + b) = 1a + a 3 b + a b + ab 3 + 1b 1 1 5th row 5 (a + b) 5 = 1a 5 + 5a b + 10a 3 b + 10a b 3 + 5ab + 1b In general, the nth row in Pascal s Triangle gives the coefficients of (a + b) n. Here are some other observations about the epansion of (a + b) n. 1. An epansion has n + 1 terms.. The power of a begins with n, decreases b 1 in each successive term, and ends with The power of b begins with 0, increases b 1 in each successive term, and ends with n.. The sum of the powers of each term is n. Using Pascal s Triangle to Epand Binomials Use Pascal s Triangle to epand (a) ( ) 5 and (b) (3 + 1) 3. a. The coefficients from the fifth row of Pascal s Triangle are 1, 5, 10, 10, 5, and 1. ( ) 5 = ( ) ( ) + 10 ( ) 3 + 5( ) + 1( ) 5 = b. The coefficients from the third row of Pascal s Triangle are 1, 3, 3, and 1. (3 + 1) 3 = 1(3) 3 + 3(3) (1) + 3(3)(1) + 1(1) 3 = Monitoring Progress Help in English and Spanish at BigIdeasMath.com 10. Use Pascal s Triangle to epand (a) (z + 3) and (b) (t 1) 5. Section. Adding, Subtracting, and Multipling Polnomials 19

17 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Describe three different methods to epand ( + 3) 3.. WRITING Is (a + b)(a b) = a b an identit? Eplain our reasoning. Monitoring Progress and Modeling with Mathematics In Eercises 3 8, find the sum. (See Eample 1.) 3. (3 + 1) + ( 3 + ). ( 5 + ) + ( ) 5. ( ) + ( ). (8 + 1) + ( ) 7. ( ) + ( ) 8. ( ) + ( ) In Eercises 9 1, find the difference. (See Eample.) 9. ( ) ( ) 10. ( ) ( ) 11. ( ) ( ) 1. ( ) ( ) 13. ( ) ( ) 1. ( ) ( ) 1. MODELING WITH MATHEMATICS A farmer plants a garden that contains corn and pumpkins. The total area (in square feet) of the garden is modeled b the epression The area of the corn is modeled b the epression 3 +. Write an epression that models the area of the pumpkins. In Eercises 17, find the product. (See Eample 3.) ( ) ( ) 19. (5 + )( + 3) 0. ( 3)( ) 1. ( )( 3 5). (3 + )( 1) 3. ( )( + 1). ( 8 )( ) ERROR ANALYSIS In Eercises 5 and, describe and correct the error in performing the operation. 15. MODELING WITH MATHEMATICS During a recent period of time, the numbers (in thousands) of males M and females F that attend degree-granting institutions in the United States can be modeled b M = 19.7t t F = 8t + 38t where t is time in ears. Write a polnomial to model the total number of people attending degree-granting institutions. Interpret its constant term. 5. ( 3 + ) ( ) = = ( 7) 3 = () = Chapter Polnomial Functions

18 In Eercises 7 3, find the product of the binomials. (See Eample.) 7. ( 3)( + )( + ) 8. ( 5)( + )( ) 9. ( )(3 + 1)( 3) 30. ( + 5)( )(3 + ) 31. (3 )(5 )( + 1) 3. ( 5)(1 )(3 + ) 33. REASONING Prove the polnomial identit (a + b)(a b) = a b. Then give an eample of two whole numbers greater than 10 that can be multiplied using mental math and the given identit. Justif our answers. (See Eample 5.) 3. NUMBER SENSE You have been asked to order tetbooks for our class. You need to order 9 tetbooks that cost $31 each. Eplain how ou can use the polnomial identit (a + b)(a b) = a b and mental math to find the total cost of the tetbooks. 9. COMPARING METHODS Find the product of the epression (a + b ) (3a b ) using two different methods. Which method do ou prefer? Eplain. 50. THOUGHT PROVOKING Adjoin one or more polgons to the rectangle to form a single new polgon whose perimeter is double that of the rectangle. Find the perimeter of the new polgon MATHEMATICAL CONNECTIONS In Eercises 51 and 5, write an epression for the volume of the figure as a polnomial in standard form. 51. V = wh 5. V = πr h MODELING WITH MATHEMATICS Two people make three deposits into their bank accounts earning the same simple interest rate r. Person A Transaction Amount 01/01/01 Deposit $ /01/013 Deposit $ /01/01 Deposit $ In Eercises 35, find the product. (See Eample.) 35. ( 9)( + 9) 3. (m + ) 37. (3c 5) 38. ( 5)( + 5) 39. (7h + ) 0. (9g ) 1. (k + ) 3. (n 3) 3 In Eercises 3 8, use Pascal s Triangle to epand the binomial. (See Eample 7.) 3. (t + ) 3. (m + ) 5. (q 3). (g + ) 5 7. (z + 1) 5 8. (np 1) Person B Transaction Amount 01/01/01 01/01/013 Deposit Deposit $ $ /01/01 Deposit $ Person A s account is worth 000(1 + r) (1 + r) (1 + r) on Januar 1, 015. a. Write a polnomial for the value of Person B s account on Januar 1, 015. b. Write the total value of the two accounts as a polnomial in standard form. Then interpret the coefficients of the polnomial. c. Suppose their interest rate is What is the total value of the two accounts on Januar 1, 015? Section. Adding, Subtracting, and Multipling Polnomials 171

19 5. PROBLEM SOLVING The sphere is centered in the cube. Find an epression for the volume of the cube outside the sphere. 55. MAKING AN ARGUMENT Your friend claims the sum of two binomials is alwas a binomial and the product of two binomials is alwas a trinomial. Is our friend correct? Eplain our reasoning. 5. HOW DO YOU SEE IT? You make a tin bo b cutting -inch-b--inch pieces of tin off the corners of a rectangle and folding up each side. The plan for our bo is shown. 1 a. What are the dimensions of the original piece of tin? b. Write a function that represents the volume of the bo. Without multipling, determine its degree ABSTRACT REASONING You are given the function f() = ( + a)( + b)( + c)( + d). When f() is written in standard form, show that the coefficient of 3 is the sum of a, b, c, and d, and the constant term is the product of a, b, c, and d. 3. DRAWING CONCLUSIONS Let g() = and h() = a. What is the degree of the polnomial g() + h()? b. What is the degree of the polnomial g() h()? c. What is the degree of the polnomial g() h()? d. In general, if g() and h() are polnomials such that g() has degree m and h() has degree n, and m > n, what are the degrees of g() + h(), g() h(), and g() h()?. FINDING A PATTERN In this eercise, ou will eplore the sequence of square numbers. The first four square numbers are represented below a. Find the differences between consecutive square numbers. Eplain what ou notice. b. Show how the polnomial identit (n + 1) n = n + 1 models the differences between square numbers. c. Prove the polnomial identit in part (b). USING TOOLS In Eercises 57 0, use a graphing calculator to make a conjecture about whether the two functions are equivalent. Eplain our reasoning. 57. f() = ( 3) 3 ; g() = h() = ( + ) 5 ; k() = f() = ( 3) ; g() = f() = ( + 5) 3 ; g() = REASONING Cop Pascal s Triangle and add rows for n =, 7, 8, 9, and 10. Use the new rows to epand ( + 3) 7 and ( 5) 9. Maintaining Mathematical Proficienc Perform the operation. Write the answer in standard form. (Section 3.). (3 i) + (5 + 9i) 7. (1 + 3i) (7 8i) 8. (7i)( 3i) 9. ( + i)( i) 5. CRITICAL THINKING Recall that a Pthagorean triple is a set of positive integers a, b, and c such that a + b = c. The numbers 3,, and 5 form a Pthagorean triple because 3 + = 5. You can use the polnomial identit ( ) + () = ( + ) to generate other Pthagorean triples. a. Prove the polnomial identit is true b showing that the simplified epressions for the left and right sides are the same. b. Use the identit to generate the Pthagorean triple when = and = 5. c. Verif that our answer in part (b) satisfies a + b = c. Reviewing what ou learned in previous grades and lessons 17 Chapter Polnomial Functions

20 .3 Dividing Polnomials Essential Question Essential Question How can ou use the factors of a cubic polnomial to solve a division problem involving the polnomial? Dividing Polnomials Work with a partner. Match each division statement with the graph of the related cubic polnomial f(). Eplain our reasoning. Use a graphing calculator to verif our answers. a. f() = ( 1)( + ) b. f() = ( 1)( + ) 1 c. e. f() = ( 1)( + ) + 1 d. f() = ( 1)( + ) + f. f() = ( 1)( + ) f() = ( 1)( + ) 3 A. B C. D. 8 8 E. F. REASONING ABSTRACTLY To be proficient in math, ou need to understand a situation abstractl and represent it smbolicall. Dividing Polnomials Work with a partner. Use the results of Eploration 1 to find each quotient. Write our answers in standard form. Check our answers b multipling. a. ( 3 + ) b. ( ) ( 1) c. ( 3 + ) ( + 1) d. ( 3 + ) ( ) e. ( ) ( + ) f. ( ) ( 3) Communicate Your Answer 3. How can ou use the factors of a cubic polnomial to solve a division problem involving the polnomial? Section.3 Dividing Polnomials 173

21 .3 Lesson What You Will Learn Core Vocabular polnomial long division, p. 17 snthetic division, p. 175 Previous long division divisor quotient remainder dividend Use long division to divide polnomials b other polnomials. Use snthetic division to divide polnomials b binomials of the form k. Use the Remainder Theorem. Long Division of Polnomials When ou divide a polnomial f() b a nonzero polnomial divisor d(), ou get a quotient polnomial q() and a remainder polnomial r(). f() d() = q() + r() d() The degree of the remainder must be less than the degree of the divisor. When the remainder is 0, the divisor divides evenl into the dividend. Also, the degree of the divisor is less than or equal to the degree of the dividend f(). One wa to divide polnomials is called polnomial long division. Using Polnomial Long Division Divide b COMMON ERROR The epression added to the quotient in the result of a long division problem is r(), not r(). d() Write polnomial division in the same format ou use when dividing numbers. Include a 0 as the coefficient of in the dividend. At each stage, divide the term with the highest power in what is left of the dividend b the first term of the divisor. This gives the net term of the quotient quotient ) = Multipl divisor b =. Subtract. Bring down net term. Multipl divisor b 33 = 3. Subtract. Bring down net term. Multipl divisor b 5 = 5. remainder Check You can check the result of a division problem b multipling the quotient b the divisor and adding the remainder. The result should be the dividend. ( 3 + 5)( ) + ( 11) = ( )( ) (3)( ) + (5)( ) 11 = = Chapter Polnomial Functions Monitoring Progress Divide using polnomial long division. Help in English and Spanish at BigIdeasMath.com 1. ( 3 + 8) ( 1). ( + + 5) ( + 1)

22 Snthetic Division There is a shortcut for dividing polnomials b binomials of the form k. This shortcut is called snthetic division. This method is shown in the net eample. Divide b 3. Using Snthetic Division Step 1 Write the coefficients of the dividend in order of descending eponents. Include a 0 for the missing -term. Because the divisor is 3, use k = 3. Write the k-value to the left of the vertical bar. k-value coefficients of Step Step 3 Bring down the leading coefficient. Multipl the leading coefficient b the k-value. Write the product under the second coefficient. Add Multipl the previous sum b the k-value. Write the product under the third coefficient. Add. Repeat this process for the remaining coefficient. The first three numbers in the bottom row are the coefficients of the quotient, and the last number is the remainder. coefficients of quotient = remainder Using Snthetic Division STUDY TIP Note that dividing polnomials does not alwas result in a polnomial. This means that the set of polnomials is not closed under division. Divide b + 1. Use snthetic division. Because the divisor is + 1 = ( 1), k = = Monitoring Progress Divide using snthetic division. Help in English and Spanish at BigIdeasMath.com 3. ( ) ( ). ( 3 7) ( + 3) Section.3 Dividing Polnomials 175

23 The Remainder Theorem The remainder in the snthetic division process has an important interpretation. When ou divide a polnomial f() b d() = k, the result is f() d() The Remainder Theorem tells ou that snthetic division can be used to evaluate a polnomial function. So, to evaluate f() when = k, divide f() b k. The remainder will be f(k). Evaluating a Polnomial Use snthetic division to evaluate f() = when =. = q() + r() d() The remainder is 359. So, ou can conclude from the Remainder Theorem that f() = 359. Check Check this b substituting = in the original function. f() = 5() 3 () + 13() + 9 = = 359 Polnomial division f() r() = q() + k k Substitute k for d(). f() = ( k)q() + r(). Multipl both sides b k. Because either r() = 0 or the degree of r() is less than the degree of k, ou know that r() is a constant function. So, let r() = r, where r is a real number, and evaluate f() when = k. f(k) = (k k)q(k) + r Substitute k for and r for r(). f(k) = r Simplif. This result is stated in the Remainder Theorem. Core Concept The Remainder Theorem If a polnomial f() is divided b k, then the remainder is r = f(k). Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use snthetic division to evaluate the function for the indicated value of. 5. f() = 10 1; = 5. f() = ; = 17 Chapter Polnomial Functions

24 .3 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Eplain the Remainder Theorem in our own words. Use an eample in our eplanation.. VOCABULARY What form must the divisor have to make snthetic division an appropriate method for dividing a polnomial? Provide eamples to support our claim. 3. VOCABULARY Write the polnomial divisor, dividend, and quotient functions represented b the snthetic division shown at the right WRITING Eplain what the colored numbers represent in the snthetic division in Eercise Monitoring Progress and Modeling with Mathematics In Eercises 5 10, divide using polnomial long division. (See Eample 1.) 5. ( + 17 ) ( ). ( ) ( 5 ) 7. ( ) ( 1 ) 8. ( ) ( + 1 ) 9. ( ) ( + ) 10. ( + 5 ) ( 3 ) In Eercises 11 18, divide using snthetic division. (See Eamples and 3.) 11. ( ) ( ) 1. ( 13 5 ) ( ) 13. ( + 7 ) ( + 5 ) 1. ( 3 + ) ( + 3 ) 15. ( + 9 ) ( 3 ) ANALYZING RELATIONSHIPS In Eercises 19, match the equivalent epressions. Justif our answers. 19. ( + 3 ) ( ) 0. ( 3 ) ( ) 1. ( + 3 ) ( ). ( ) ( ) 1 A C B D ERROR ANALYSIS In Eercises 3 and, describe and correct the error in using snthetic division to divide b = ( ) ( 1 ) 17. ( ) ( ) 18. ( ) ( + 5 ) = 3 3 Section.3 Dividing Polnomials 177

25 In Eercises 5 3, use snthetic division to evaluate the function for the indicated value of. (See Eample.) 5. f() = ; = 1. f() = 3 + 0; = 3 7. f() = ; = 8. f() = ; = 9. f() = 3 + 1; = 30. f() = 3 9 7; = f() = ; = 3 3. f() = 3 ; = 5 3. COMPARING METHODS The profit P (in millions of dollars) for a DVD manufacturer can be modeled b P = 3 + 7, where is the number (in millions) of DVDs produced. Use snthetic division to show that the compan ields a profit of $9 million when million DVDs are produced. Is there an easier method? Eplain. 37. CRITICAL THINKING What is the value of k such that ( 3 + k 30 ) ( 5 ) has a remainder of zero? A 1 B C D HOW DO YOU SEE IT? The graph represents the polnomial function f() = MAKING AN ARGUMENT You use snthetic division to divide f() b ( a) and find that the remainder equals 15. Your friend concludes that f(15) = a. Is our friend correct? Eplain our reasoning THOUGHT PROVOKING A polgon has an area represented b A = The figure has at least one dimension equal to +. Draw the figure and label its dimensions. 35. USING TOOLS The total attendance A (in thousands) at NCAA women s basketball games and the number T of NCAA women s basketball teams over a period of time can be modeled b A = T = where is in ears and 0 < < 18. Write a function for the average attendance per team over this period of time a. The epression f() ( k) has a remainder of 15. What is the value of k? b. Use the graph to compare the remainders of ( ) ( + 3 ) and ( ) ( + 1 ). 39. MATHEMATICAL CONNECTIONS The volume V of the rectangular prism is given b V = Find an epression for the missing dimension. +? + Maintaining Mathematical Proficienc Find the zero(s) of the function. (Sections 3.1 and 3.) 1. f() = + 9. g() = 3( + )( ) 3. g() = h() = USING STRUCTURE You divide two polnomials and obtain the result What is the + dividend? How did ou find it? Reviewing what ou learned in previous grades and lessons 178 Chapter Polnomial Functions

26 . Factoring Polnomials Essential Question How can ou factor a polnomial? Factoring Polnomials Work with a partner. Match each polnomial equation with the graph of its related polnomial function. Use the -intercepts of the graph to write each polnomial in factored form. Eplain our reasoning. a = 0 b. 3 + = 0 c. 3 + = 0 d. 3 = 0 e. 5 + = 0 f. 3 + = 0 A. B. C. D. E. F. MAKING SENSE OF PROBLEMS To be proficient in math, ou need to check our answers to problems and continuall ask ourself, Does this make sense? Factoring Polnomials Work with a partner. Use the -intercepts of the graph of the polnomial function to write each polnomial in factored form. Eplain our reasoning. Check our answers b multipling. a. f() = b. f() = 3 c. f() = 3 3 d. f() = e. f() = + 3 f. f() = Communicate Your Answer 3. How can ou factor a polnomial?. What information can ou obtain about the graph of a polnomial function written in factored form? Section. Factoring Polnomials 179

27 . Lesson What You Will Learn Core Vocabular factored completel, p. 180 factor b grouping, p. 181 quadratic form, p. 181 Previous zero of a function snthetic division Factor polnomials. Use the Factor Theorem. Factoring Polnomials Previousl, ou factored quadratic polnomials. You can also factor polnomials with degree greater than. Some of these polnomials can be factored completel using techniques ou have previousl learned. A factorable polnomial with integer coefficients is factored completel when it is written as a product of unfactorable polnomials with integer coefficients. Finding a Common Monomial Factor Factor each polnomial completel. a. 3 5 b c. 5z + 30z 3 + 5z a. 3 5 = ( 5) Factor common monomial. = ( 5)( + 1) Factor trinomial. b = 3 3 ( 1) Factor common monomial. = 3 3 ( )( + ) Difference of Two Squares Pattern c. 5z + 30z 3 + 5z = 5z (z + z + 9) Factor common monomial. = 5z (z + 3) Perfect Square Trinomial Pattern Monitoring Progress Help in English and Spanish at BigIdeasMath.com Factor the polnomial completel n 7 75n m 5 1m + 8m 3 In part (b) of Eample 1, the special factoring pattern for the difference of two squares was used to factor the epression completel. There are also factoring patterns that ou can use to factor the sum or difference of two cubes. Core Concept Special Factoring Patterns Sum of Two Cubes Eample a 3 + b 3 = (a + b)(a ab + b ) = () = ( + 1)(1 + 1) Difference of Two Cubes Eample a 3 b 3 = (a b)(a + ab + b ) = (3) 3 3 = (3 )(9 + + ) 180 Chapter Polnomial Functions

28 Factoring the Sum or Difference of Two Cubes Factor (a) 3 15 and (b) 1s 5 + 5s completel. a = Write as a 3 b 3. = ( 5)( ) Difference of Two Cubes Pattern b. 1s 5 + 5s = s (8s 3 + 7) Factor common monomial. = s [(s) ] Write 8s as a 3 + b 3. = s (s + 3)(s s + 9) Sum of Two Cubes Pattern For some polnomials, ou can factor b grouping pairs of terms that have a common monomial factor. The pattern for factoring b grouping is shown below. ra + rb + sa + sb = r(a + b) + s(a + b) = (r + s)(a + b) Factoring b Grouping Factor z 3 + 5z z 0 completel. z 3 + 5z z 0 = z (z + 5) (z + 5) Factor b grouping. = (z )(z + 5) Distributive Propert = (z )(z + )(z + 5) Difference of Two Squares Pattern LOOKING FOR STRUCTURE The epression 1 81 is in quadratic form because it can be written as u 81 where u =. An epression of the form au + bu + c, where u is an algebraic epression, is said to be in quadratic form. The factoring techniques ou have studied can sometimes be used to factor such epressions. Factoring Polnomials in Quadratic Form Factor (a) 1 81 and (b) 3p p p completel. a = ( ) 9 Write as a b. = ( + 9)( 9) Difference of Two Squares Pattern = ( + 9)( 3)( + 3) Difference of Two Squares Pattern b. 3p p p = 3p ( p + 5p 3 + ) Factor common monomial. = 3p ( p 3 + 3)( p 3 + ) Factor trinomial in quadratic form. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Factor the polnomial completel.. a z 5 750z n w 5w + 30w Section. Factoring Polnomials 181

29 The Factor Theorem When dividing polnomials in the previous section, the eamples had nonzero remainders. Suppose the remainder is 0 when a polnomial f() is divided b k. Then, f() k = q() + 0 k = q() where q() is the quotient polnomial. Therefore, f() = ( k) q(), so that k is a factor of f(). This result is summarized b the Factor Theorem, which is a special case of the Remainder Theorem. READING In other words, k is a factor of f () if and onl if k is a zero of f. Core Concept The Factor Theorem A polnomial f() has a factor k if and onl if f(k) = 0. STUDY TIP In part (b), notice that direct substitution would have resulted in more difficult computations than snthetic division. Determining Whether a Linear Binomial Is a Factor Determine whether (a) is a factor of f() = + and (b) + 5 is a factor of f() = a. Find f() b direct substitution. b. Find f( 5) b snthetic division. f() = + () = = Because f() 0, the binomial Because f( 5) = 0, the binomial is not a factor of + 5 is a factor of f() = +. f() = Factoring a Polnomial Show that + 3 is a factor of f() = Then factor f() completel. ANOTHER WAY Notice that ou can factor f () b grouping. f () = 3 ( + 3) 1( + 3) = ( 3 1)( + 3) = ( + 3)( 1) ( + + 1) Show that f( 3) = 0 b snthetic division Because f( 3) = 0, ou can conclude that + 3 is a factor of f() b the Factor Theorem. Use the result to write f() as a product of two factors and then factor completel. f() = Write original polnomial. = ( + 3)( 3 1) Write as a product of two factors. = ( + 3)( 1)( + + 1) Difference of Two Cubes Pattern 18 Chapter Polnomial Functions

30 Because the -intercepts of the graph of a function are the zeros of the function, ou can use the graph to approimate the zeros. You can check the approimations using the Factor Theorem. STUDY TIP You could also check that is a zero using the original function, but using the quotient polnomial helps ou find the remaining factor. Real-Life Application During the first 5 seconds of a roller coaster ride, the function h(t) = t 3 1t + 9t + 3 represents the height h (in feet) of the roller coaster after t seconds. How long is the roller coaster at or below ground level in the first 5 seconds? 1. Understand the Problem You are given a function rule that represents the height of a roller coaster. You are asked to determine how long the roller coaster is at or below ground during the first 5 seconds of the ride.. Make a Plan Use a graph to estimate the zeros of the function and check using the Factor Theorem. Then use the zeros to describe where the graph lies below the t-ais. 3. Solve the Problem From the graph, two of the zeros appear to be 1 and. The third zero is between and 5. Step 1 Determine whether 1 is a zero using snthetic division h( 1) = 0, so 1 is a zero of h and t + 1 is a factor of h(t). Step Determine whether is a zero. If is also a zero, then t is a factor of the resulting quotient polnomial. Check using snthetic division So, h(t) = (t + 1)(t )(t 17). The factor t 17 indicates that the zero between and 5 is 17, or.5. The zeros are 1,, and.5. Onl t = and t =.5 occur in the first 5 seconds. The graph shows that the roller coaster is at or below ground level for.5 =.5 seconds.. Look Back Use a table of values to verif the positive zeros and heights between the zeros. zero zero 80 0 The remainder is 0, so t is a factor of h(t) and is a zero of h. X X= h(t) = t 3 1t + 9t + 3 Y h 1 5 t negative Monitoring Progress Help in English and Spanish at BigIdeasMath.com 10. Determine whether is a factor of f() = Show that is a factor of f() = 3 5. Then factor f() completel. 1. In Eample 7, does our answer change when ou first determine whether is a zero and then whether 1 is a zero? Justif our answer. Section. Factoring Polnomials 183

31 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The epression 9 9 is in form because it can be written as u 9 where u =.. VOCABULARY Eplain when ou should tr factoring a polnomial b grouping. 3. WRITING How do ou know when a polnomial is factored completel?. WRITING Eplain the Factor Theorem and wh it is useful. Monitoring Progress and Modeling with Mathematics In Eercises 5 1, factor the polnomial completel. (See Eample 1.) k 5 100k p 5 19p 3 8. m m 5 + m 9. q + 9q 3 18q 10. 3r 11r 5 0r w 10 19w 9 + w v v 8 + 1v 7 In Eercises 13 0, factor the polnomial completel. (See Eample.) g c h 9 19h 18. 9n 51n t t z z 8 ERROR ANALYSIS In Eercises 1 and, describe and correct the error in factoring the polnomial = 3( + 9) = 3( + 3)( 3) In Eercises 3 30, factor the polnomial completel. (See Eample 3.) m 3 m + 7m a a + 8a + 8. k 3 0k + 5k z 3 5z 9z q 3 1q 9q n 3 + 3n n In Eercises 31 38, factor the polnomial completel. (See Eample.) 31. 9k 9 3. m c + 9c z a r 8 + 3r 5 0r 38. n 1 3n 7 + 8n In Eercises 39, determine whether the binomial is a factor of the polnomial. (See Eample 5.) 39. f() = ; 0. g() = ; = ( 3 ) 3 + () 3 = ( 3 + )[( 3 ) ( 3 )() + () ] = ( 3 + )( + ) 1. h() = ; + 3. g() = ; 3. h() = ; + 18 Chapter Polnomial Functions. t() = ; +

32 In Eercises 5 50, show that the binomial is a factor of the polnomial. Then factor the polnomial completel. (See Eample.) 5. g() = 3 0; +. t() = ; 5 7. f() = ; 5. MODELING WITH MATHEMATICS The volume (in cubic inches) of a rectangular birdcage can be modeled b V = , where is the length (in inches). Determine the values of for which the model makes sense. Eplain our reasoning. V 8. s() = + 3 5; + 9. r() = ; h() = 3 3; + ANALYZING RELATIONSHIPS In Eercises 51 5, match the function with the correct graph. Eplain our reasoning. 51. f() = ( )( 3)( + 1) 5. g() = ( + )( + 1)( ) 53. h() = ( + )( + 3)( 1) 5. k() = ( )( 1)( + ) USING STRUCTURE In Eercises 57, use the method of our choice to factor the polnomial completel. Eplain our reasoning. 57. a + a 5 30a 58. 8m z 3 7z 9z p 8 1p 5 + 1p 1. r n 1. 9k 3 k + 3k 8 A. B. 5. REASONING Determine whether each polnomial is factored completel. If not, factor completel. a. 7z (z z ) b. ( n)(n + n)(3n 11) c. 3( 5)(9 ) C. 55. MODELING WITH MATHEMATICS The volume (in cubic inches) of a shipping bo is modeled b V = , where is the length (in inches). Determine the values of for which the model makes sense. Eplain our reasoning. (See Eample 7.) 0 0 V D. 8. PROBLEM SOLVING The profit P (in millions of dollars) for a T-shirt manufacturer can be modeled b P = 3 + +, where is the number (in millions) of T-shirts produced. Currentl the compan produces million T-shirts and makes a profit of $ million. What lesser number of T-shirts could the compan produce and still make the same profit? 7. PROBLEM SOLVING The profit P (in millions of dollars) for a shoe manufacturer can be modeled b P = 1 3 +, where is the number (in millions) of shoes produced. The compan now produces 1 million shoes and makes a profit of $5 million, but it would like to cut back production. What lesser number of shoes could the compan produce and still make the same profit? Section. Factoring Polnomials 185

33 8. THOUGHT PROVOKING Find a value of k such that f() has a remainder of 0. Justif our answer. k f() = COMPARING METHODS You are taking a test where calculators are not permitted. One question asks ou to evaluate g(7) for the function g() = You use the Factor Theorem and snthetic division and our friend uses direct substitution. Whose method do ou prefer? Eplain our reasoning. 70. MAKING AN ARGUMENT You divide f() b ( a) and find that the remainder does not equal 0. Your friend concludes that f() cannot be factored. Is our friend correct? Eplain our reasoning. 71. CRITICAL THINKING What is the value of k such that 7 is a factor of h() = 3 13 k + 105? Justif our answer. 7. HOW DO YOU SEE IT? Use the graph to write an equation of the cubic function in factored form. Eplain our reasoning. 7. REASONING The graph of the function f() = is shown. Can ou use the Factor Theorem to factor f()? Eplain. 75. MATHEMATICAL CONNECTIONS The standard equation of a circle with radius r and center (h, k) is ( h) + ( k) = r. Rewrite each equation of a circle in standard form. Identif the center and radius of the circle. Then graph the circle. r (h, k) a = 5 b. + + = 9 (, ) c = ABSTRACT REASONING Factor each polnomial completel. a. 7ac + bc 7ad bd b. n n + 1 c. a 5 b a b + a b ab 3 + a 3 b Maintaining Mathematical Proficienc Solve the quadratic equation b factoring. (Section 3.1) = = = = 0 Solve the quadratic equation b completing the square. (Section 3.3) = = = = 0 7. CRITICAL THINKING Use the diagram to complete parts (a) (c). a. Eplain wh a 3 b 3 is equal to the sum of the volumes of the solids I, II, and III. b. Write an algebraic epression for the volume of each of the three solids. Leave II III our epressions in a b factored form. b b c. Use the results from I part (a) and part (b) a to derive the factoring a pattern a 3 b 3. Reviewing what ou learned in previous grades and lessons 18 Chapter Polnomial Functions

34 .1. What Did You Learn? Core Vocabular polnomial, p. 158 polnomial function, p. 158 end behavior, p. 159 Pascal s Triangle, p. 19 polnomial long division, p. 17 snthetic division, p. 175 factored completel, p. 180 factor b grouping, p. 181 quadratic form, p. 181 Core Concepts Section.1 Common Polnomial Functions, p. 158 End Behavior of Polnomial Functions, p. 159 Section. Operations with Polnomials, p. 1 Special Product Patterns, p. 17 Section.3 Polnomial Long Division, p. 17 Snthetic Division, p. 175 Section. Factoring Polnomials, p. 180 Special Factoring Patterns, p. 180 Graphing Polnomial Functions, p. 10 Pascal s Triangle, p. 19 The Remainder Theorem, p. 17 The Factor Theorem, p. 18 Mathematical Practices 1. Describe the entr points ou used to analze the function in Eercise 3 on page 1.. Describe how ou maintained oversight in the process of factoring the polnomial in Eercise 9 on page 185. Stud Skills Keeping Your Mind Focused When ou sit down at our desk, review our notes from the last class. Repeat in our mind what ou are writing in our notes. When a mathematical concept is particularl difficult, ask our teacher for another eample. 187

35 .1. Quiz Decide whether the function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient. (Section.1) 1. f() = g() = h() = Describe the -values for which (a) f is increasing or decreasing, (b) f() > 0, and (c) f() < 0. (Section.1) (, 3) (3, 0) (1, 0) f 5. Write an epression for the area and perimeter for the figure shown. (Section.) Perform the indicated operation. (Section.). (7 ) ( ) 7. ( 3 + )(3 1) 8. ( 1)( + 3)( ) 9. Use Pascal s Triangle to epand ( + ) 5. (Section.) 10. Divide b 1. (Section.3) Factor the polnomial completel. (Section.) 11. a 3 a 8a 1. 8m z 3 + z z 1. 9b 15. Show that + 5 is a factor of f() = Then factor f() completel. (Section.) 1. The estimated price P (in cents) of stamps in the United States can be modeled b the polnomial function P(t) = 0.007t 3 0.1t + 1t + 17, where t represents the number of ears since (Section.1) a. Use a graphing calculator to graph the function for the interval 0 t 0. Describe the behavior of the graph on this interval. b. What was the average rate of change in the price of stamps from 1990 to 010? 17. The volume V (in cubic feet) of a rectangular wooden crate is modeled b the function V() = , where is the width (in feet) of the crate. Determine the values of for which the model makes sense. Eplain our reasoning. (Section.) V() = V 188 Chapter Polnomial Functions

36 .5 Solving Polnomial Equations Essential Question Essential Question How can ou determine whether a polnomial equation has a repeated solution? Cubic Equations and Repeated Solutions USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. Work with a partner. Some cubic equations have three distinct solutions. Others have repeated solutions. Match each cubic polnomial equation with the graph of its related polnomial function. Then solve each equation. For those equations that have repeated solutions, describe the behavior of the related function near the repeated zero using the graph or a table of values. a = 0 b = 0 c = 0 d. 3 + = 0 e. 3 3 = 0 f = 0 A. B. C. D. E. F. Quartic Equations and Repeated Solutions Work with a partner. Determine whether each quartic equation has repeated solutions using the graph of the related quartic function or a table of values. Eplain our reasoning. Then solve each equation. a = 0 b. 3 + = 0 c. 3 + = 0 d = 0 Communicate Your Answer 3. How can ou determine whether a polnomial equation has a repeated solution?. Write a cubic or a quartic polnomial equation that is different from the equations in Eplorations 1 and and has a repeated solution. Section.5 Solving Polnomial Equations 189

37 .5 Lesson What You Will Learn Find solutions of polnomial equations and zeros of polnomial functions. Core Vocabular repeated solution, p. 190 Previous roots of an equation real numbers conjugates Use the Rational Root Theorem. Use the Irrational Conjugates Theorem. Finding Solutions and Zeros You have used the Zero-Product Propert to solve factorable quadratic equations. You can etend this technique to solve some higher-degree polnomial equations. Solve = 0. Solving a Polnomial Equation b Factoring Check = 0 Write the equation. ( + 9) = 0 Factor common monomial. Zero X=3 Y=0 ( 3) = 0 Perfect Square Trinomial Pattern = 0 or ( 3) = 0 Zero-Product Propert = 0 or = 3 Solve for. The solutions, or roots, are = 0 and = 3. STUDY TIP Because the factor 3 appears twice, the root = 3 has a multiplicit of. In Eample 1, the factor 3 appears more than once. This creates a repeated solution of = 3. Note that the graph of the related function touches the -ais (but does not cross the -ais) at the repeated zero = 3, and crosses the -ais at the zero = 0. This concept can be generalized as follows. When a factor k of f() is raised to an odd power, the graph of f crosses the -ais at = k. When a factor k of f() is raised to an even power, the graph of f touches the -ais (but does not cross the -ais) at = k. Finding Zeros of a Polnomial Function Find the zeros of f() = Then sketch a graph of the function. (, 0) (, 0) 0 = Set f() equal to 0. 0 = ( 8 + 1) Factor out. 0 = ( )( ) Factor trinomial in quadratic form. 0 = ( + )( )( + )( ) Difference of Two Squares Pattern 0 = ( + ) ( ) Rewrite using eponents. 0 (0, 3) Because both factors + and are raised to an even power, the graph of f touches the -ais at the zeros = and =. B analzing the original function, ou can determine that the -intercept is 3. Because the degree is even and the leading coefficient is negative, f() as and f() as +. Use these characteristics to sketch a graph of the function. 190 Chapter Polnomial Functions

38 Monitoring Progress Solve the equation. Help in English and Spanish at BigIdeasMath.com = = 1 3 Find the zeros of the function. Then sketch a graph of the function. 3. f() = f() = 3 + STUDY TIP Notice that ou can use the Rational Root Theorem to list possible zeros of polnomial functions. The Rational Root Theorem The solutions of the equation = 0 are, 3, and 7 8. Notice that the numerators (5, 3, and 7) of the zeros are factors of the constant term, 105. Also notice that the denominators (,, and 8) are factors of the leading coefficient,. These observations are generalized b the Rational Root Theorem. Core Concept The Rational Root Theorem If f() = a n n + + a 1 + a 0 has integer coefficients, then ever rational solution of f() = 0 has the following form: p q = factor of constant term a 0 factor of leading coefficient a n The Rational Root Theorem can be a starting point for finding solutions of polnomial equations. However, the theorem lists onl possible solutions. In order to find the actual solutions, ou must test values from the list of possible solutions. 5 Using the Rational Root Theorem Find all real solutions of = 0. ANOTHER WAY You can use direct substitution to test possible solutions, but snthetic division helps ou identif other factors of the polnomial. The polnomial f() = is not easil factorable. Begin b using the Rational Root Theorem. Step 1 List the possible rational solutions. The leading coefficient of f() is 1 and the constant term is 0. So, the possible rational solutions of f() = 0 are 1 = ± 1, ± 1, ± 1, ± 5 1, ± 10 1, ± 0 1. Step Test possible solutions using snthetic division until a solution is found. Test = 1: Test = 1: f(1) 0, so 1 is not a factor of f(). Step 3 Factor completel using the result of the snthetic division. ( + 1)( 9 + 0) = 0 Write as a product of factors. ( + 1)( )( 5) = 0 Factor the trinomial. So, the solutions are = 1, =, and = 5. f( 1) = 0, so + 1 is a factor of f(). Section.5 Solving Polnomial Equations 191

39 In Eample 3, the leading coefficient of the polnomial is 1. When the leading coefficient is not 1, the list of possible rational solutions or zeros can increase dramaticall. In such cases, the search can be shortened b using a graph. Finding Zeros of a Polnomial Function Find all real zeros of f() = Step 1 List the possible rational zeros of f : ± 1, ± 1, ± 3 1, ± 1, ± 1, ± 1 1, 1 ±, ± 3, ± 1 5, ± 5, ± 3 5, ± 5, ± 5, ± 1 5, ± 1 10, ± 3 10 Step Choose reasonable values from the list above to test using the graph of the function. For f, the values 3 =, = 1, = 3 5, and = are reasonable based on the graph shown at the right f 5 Step 3 Test the values using snthetic division until a zero is found Step Factor out a binomial using the result of the snthetic division. 1 is a zero. f() = ( + ) 1 ( ) Write as a product of factors. = ( + ) ) Factor out of the second factor. = ( + 1)( ) Multipl the first factor b. Step 5 Repeat the steps above for g() = An zero of g will also be a zero of f. The possible rational zeros of g are: 5 g 1 = ±1, ±, ±3, ±, ±, ±1, ± 5, ± 5, ± 3 5, ± 5, ± 5, ± The graph of g shows that 3 5 ma be a zero. Snthetic division shows that 3 5 is a zero and g() = ( 5) 3 (5 5 0) = (5 3)( ). It follows that: f() = ( + 1) g() = ( + 1)(5 3)( ) Step Find the remaining zeros of f b solving = 0. = ( 1) ± ( 1) (1)() (1) Substitute 1 for a, 1 for b, and for c in the Quadratic Formula. = 1 ± 17 The real zeros of f are 1, 3 5, Simplif..5, and Chapter Polnomial Functions

40 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Find all real solutions of = 0.. Find all real zeros of f() = The Irrational Conjugates Theorem In Eample, notice that the irrational zeros are conjugates of the form a + b and a b. This illustrates the theorem below. Core Concept The Irrational Conjugates Theorem Let f be a polnomial function with rational coefficients, and let a and b be rational numbers such that b is irrational. If a + b is a zero of f, then a b is also a zero of f. Using Zeros to Write a Polnomial Function Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the zeros 3 and + 5. Because the coefficients are rational and + 5 is a zero, 5 must also be a zero b the Irrational Conjugates Theorem. Use the three zeros and the Factor Theorem to write f() as a product of three factors. f() = ( 3) [ ( + 5 )][ ( 5 )] Write f() in factored form. = ( 3) [ ( ) 5 ] [ ( ) + 5 ] Regroup terms. = ( 3) [ ( ) 5 ] Multipl. = ( 3) [ ( + ) 5 ] Epand binomial. = ( 3)( 1) Simplif. = Multipl. = Combine like terms. Check You can check this result b evaluating f at each of its three zeros. f(3) = 3 3 7(3) + 11(3) + 3 = = 0 f ( + 5 ) = ( + 5 ) 3 7 ( + 5 ) + 11 ( + 5 ) + 3 = = 0 Because f ( + 5 ) = 0, b the Irrational Conjugates Theorem f ( 5 ) = 0. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 7. Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the zeros and 1 5. Section.5 Solving Polnomial Equations 193

41 .5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE If a polnomial function f has integer coefficients, then ever rational solution of f() = 0 has the form p, where p is a factor of the and q is a factor of q the.. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. Find the -intercept of the graph of = 3 +. Find all the real solutions of 3 + = 0. Find the -intercepts of the graph of = 3 +. Find the real zeros of f() = 3 +. Monitoring Progress and Modeling with Mathematics In Eercises 3 1, solve the equation. (See Eample 1.) 3. z 3 z 1z = 0. a 3 a + a = =. v 3 v 1v = w 3 = 50w 8. 9m 5 = 7m 3 9. c c 3 = 1c 3c 10. p + 0 = 1p 11. 1n + 8n = n = 9 7 In Eercises 13 0, find the zeros of the function. Then sketch a graph of the function. (See Eample.) 13. h() = f() = p() = g() = g() = h() = h() = p() = USING EQUATIONS According to the Rational Root Theorem, which is not a possible solution of the equation = 0? A 9 B C 1 5 D 3. USING EQUATIONS According to the Rational Root Theorem, which is not a possible zero of the function f() = ? A 3 B 8 C 3 3 D ERROR ANALYSIS In Eercises 3 and, describe and correct the error in listing the possible rational zeros of the function. 3.. f () = Possible rational zeros of f : 1, 3, 5, 9, 15, 5 f () = Possible rational zeros of f : 1 ±1, ±3, ±, ± 1, ± 1 8, ± 3, ± 3, ± 3 8 In Eercises 5 3, find all the real solutions of the equation. (See Eample 3.) = = Chapter Polnomial Functions

42 = = = = = = 0 In Eercises 33 38, find all the real zeros of the function. (See Eample.) 33. f() = g() = h() = f() = p() = g() = USING TOOLS In Eercises 39 and 0, use the graph to shorten the list of possible rational zeros of the function. Then find all real zeros of the function. 39. f() = f() = In Eercises 1, write a polnomial function f of least degree that has a leading coefficient of 1 and the given zeros. (See Eample 5.) 1., 3,.,, 5 3., , 7 5., 0, , 5, COMPARING METHODS Solve the equation = 0 using two different methods. Which method do ou prefer? Eplain our reasoning. 9. PROBLEM SOLVING At a factor, molten glass is poured into molds to make paperweights. Each mold is a rectangular prism with a height 3 centimeters greater than the length of each side of its square base. Each mold holds 11 cubic centimeters of glass. What are the dimensions of the mold? 50. MATHEMATICAL CONNECTIONS The volume of the cube shown is 8 cubic centimeters. a. Write a polnomial equation that ou can 3 use to find the value of. b. Identif the possible 3 rational solutions of the 3 equation in part (a). c. Use snthetic division to find a rational solution of the equation. Show that no other real solutions eist. d. What are the dimensions of the cube? 51. PROBLEM SOLVING Archaeologists discovered a huge hdraulic concrete block at the ruins of Caesarea with a volume of 95 cubic meters. The block is meters high b 1 15 meters long b 1 1 meters wide. What are the dimensions of the block? 5. MAKING AN ARGUMENT Your friend claims that when a polnomial function has a leading coefficient of 1 and the coefficients are all integers, ever possible rational zero is an integer. Is our friend correct? Eplain our reasoning. 53. MODELING WITH MATHEMATICS During a 10-ear period, the amount (in millions of dollars) of athletic equipment E sold domesticall can be modeled b E(t) = 0t 3 + 5t 80t + 1,1, where t is in ears. a. Write a polnomial equation to find the ear when about $,01,000,000 of athletic equipment is sold. b. List the possible whole-number solutions of the equation in part (a). Consider the domain when making our list of possible solutions. c. Use snthetic division to find when $,01,000,000 of athletic equipment is sold. 8. REASONING Is it possible for a cubic function to have more than three real zeros? Eplain. Section.5 Solving Polnomial Equations 195

43 5. THOUGHT PROVOKING Write a third or fourth degree polnomial function that has zeros at ± 3. Justif our answer. 55. MODELING WITH MATHEMATICS You are designing a marble basin that will hold a fountain for a cit park. The sides and bottom of the basin should be 1 foot thick. Its outer length should be twice its outer width and outer height. What should the outer dimensions of the basin be if it is to hold 3 cubic feet of water? 1 ft 5. HOW DO YOU SEE IT? Use the information in the graph to answer the questions. 58. WRITING EQUATIONS Write a polnomial function g of least degree that has rational coefficients, a leading coefficient of 1, and the zeros + 7 and 3 +. In Eercises 59, solve f() = g() b graphing and algebraic methods. 59. f() = 3 + 1; g() = f() = ; g() = f() = 3 + ; g() = +. f() = ; g() = 9 3. MODELING WITH MATHEMATICS You are building a pair of ramps for a loading platform. The left ramp is twice as long as the right ramp. If 150 cubic feet of concrete are used to build the ramps, what are the dimensions of each ramp? f a. What are the real zeros of the function f? b. Write an equation of the quartic function in factored form. 57. REASONING Determine the value of k for each equation so that the given -value is a solution. a k = 0; = b k 18 = 0; = c. k = 0; = 5 Maintaining Mathematical Proficienc. MODELING WITH MATHEMATICS Some ice sculptures are made b filling a mold and then freezing it. You are making an ice mold for a school dance. It is to be shaped like a pramid with a height 1 foot greater than the length of each side of its + 1 square base. The volume of the ice sculpture is cubic feet. What are the dimensions of the mold? 5. ABSTRACT REASONING Let a n be the leading coefficient of a polnomial function f and a 0 be the constant term. If a n has r factors and a 0 has s factors, what is the greatest number of possible rational zeros of f that can be generated b the Rational Zero Theorem? Eplain our reasoning. Reviewing what ou learned in previous grades and lessons Decide whether the function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient. (Section.1). h() = g() = f() = p() = Find the zeros of the function. (Section 3.) 70. f() = g() = h() = f() = Chapter Polnomial Functions

44 . The Fundamental Theorem of Algebra Essential Question Essential Question How can ou determine whether a polnomial equation has imaginar solutions? Cubic Equations and Imaginar Solutions Work with a partner. Match each cubic polnomial equation with the graph of its related polnomial function. Then find all solutions. Make a conjecture about how ou can use a graph or table of values to determine the number and tpes of solutions of a cubic polnomial equation. a = 0 b. 3 + = 0 c. 3 + = 0 d = 0 e = 0 f = 0 A. B. C. D. E. F. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technolog to enable ou to visualize results and eplore consequences. Quartic Equations and Imaginar Solutions Work with a partner. Use the graph of the related quartic function, or a table of values, to determine whether each quartic equation has imaginar solutions. Eplain our reasoning. Then find all solutions. a. 3 + = 0 b. 1 = 0 c = 0 d = 0 Communicate Your Answer 3. How can ou determine whether a polnomial equation has imaginar solutions?. Is it possible for a cubic equation to have three imaginar solutions? Eplain our reasoning. Section. The Fundamental Theorem of Algebra 197

45 . Lesson What You Will Learn Core Vocabular comple conjugates, p. 199 Previous repeated solution degree of a polnomial solution of an equation zero of a function conjugates Use the Fundamental Theorem of Algebra. Find conjugate pairs of comple zeros of polnomial functions. Use Descartes s Rule of Signs. The Fundamental Theorem of Algebra The table shows several polnomial equations and their solutions, including repeated solutions. Notice that for the last equation, the repeated solution = 1 is counted twice. Equation Degree Solution(s) Number of solutions 1 = = 0 ± 3 8 = 0 3, 1 ± i = 0 3 1, 1, 1 3 In the table, note the relationship between the degree of the polnomial f() and the number of solutions of f() = 0. This relationship is generalized b the Fundamental Theorem of Algebra, first proven b German mathematician Carl Friedrich Gauss ( ). STUDY TIP The statements the polnomial equation f () = 0 has eactl n solutions and the polnomial function f has eactl n zeros are equivalent. Core Concept The Fundamental Theorem of Algebra Theorem If f() is a polnomial of degree n where n > 0, then the equation f() = 0 has at least one solution in the set of comple numbers. Corollar If f() is a polnomial of degree n where n > 0, then the equation f() = 0 has eactl n solutions provided each solution repeated twice is counted as two solutions, each solution repeated three times is counted as three solutions, and so on. The corollar to the Fundamental Theorem of Algebra also means that an nth-degree polnomial function f has eactl n zeros. Finding the Number of Solutions or Zeros a. How man solutions does the equation = 0 have? b. How man zeros does the function f() = have? a. Because = 0 is a polnomial equation of degree 3, it has three solutions. (The solutions are 3, i, and i.) b. Because f() = is a polnomial function of degree, it has four zeros. (The zeros are,,, and 0.) 198 Chapter Polnomial Functions

46 Finding the Zeros of a Polnomial Function Find all zeros of f() = STUDY TIP Notice that ou can use imaginar numbers to write ( + ) as ( + i )( i ). In general, (a + b ) = (a + bi )(a bi ). Step 1 Find the rational zeros of f. Because f is a polnomial function of degree 5, it has five zeros. The possible rational zeros are ±1, ±, ±, and ±8. Using snthetic division, ou can determine that 1 is a zero repeated twice and is also a zero. Step Write f() in factored form. Dividing f() b its known factors + 1, + 1, and gives a quotient of +. So, f() = ( + 1) ( )( + ). Step 3 Find the comple zeros of f. Solving + = 0, ou get = ±i. This means + = ( + i )( i ). f() = ( + 1) ( )( + i )( i ) From the factorization, there are five zeros. The zeros of f are 1, 1,, i, and i. The graph of f and the real zeros are shown. Notice that onl the real zeros appear as -intercepts. Also, the graph of f touches the -ais at the repeated zero = 1 and crosses the -ais at = Zero X=-1 Y=0 5 Zero X= Y=0 5 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. How man solutions does the equation = 0 have?. How man zeros does the function f() = have? Find all zeros of the polnomial function. 3. f() = f() = Comple Conjugates Pairs of comple numbers of the forms a + bi and a bi, where b 0, are called comple conjugates. In Eample, notice that the zeros i and i are comple conjugates. This illustrates the net theorem. Core Concept The Comple Conjugates Theorem If f is a polnomial function with real coefficients, and a + bi is an imaginar zero of f, then a bi is also a zero of f. Section. The Fundamental Theorem of Algebra 199

47 Using Zeros to Write a Polnomial Function Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the zeros and 3 + i. Because the coefficients are rational and 3 + i is a zero, 3 i must also be a zero b the Comple Conjugates Theorem. Use the three zeros and the Factor Theorem to write f() as a product of three factors. f() = ( )[ (3 + i)][ (3 i)] Write f() in factored form. = ( )[( 3) i][( 3) + i] Regroup terms. = ( )[( 3) i ] Multipl. = ( )[( + 9) ( 1)] Epand binomial and use i = 1. = ( )( + 10) Simplif. = Multipl. = Combine like terms. Check You can check this result b evaluating f at each of its three zeros. f() = () 3 8() + () 0 = = 0 f(3 + i) = (3 + i) 3 8(3 + i) + (3 + i) 0 = 18 + i 8i + + i 0 = 0 Because f(3 + i) = 0, b the Comple Conjugates Theorem f(3 i) = 0. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. 5. 1, i. 3, 1 + i 5 7., 1 3i 8., i, Descartes s Rule of Signs French mathematician René Descartes ( ) found the following relationship between the coefficients of a polnomial function and the number of positive and negative zeros of the function. Core Concept Descartes s Rule of Signs Let f() = a n n + a n 1 n a + a 1 + a 0 be a polnomial function with real coefficients. The number of positive real zeros of f is equal to the number of changes in sign of the coefficients of f() or is less than this b an even number. The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f( ) or is less than this b an even number. 00 Chapter Polnomial Functions

48 Using Descartes s Rule of Signs Determine the possible numbers of positive real zeros, negative real zeros, and imaginar zeros for f() = f() = The coefficients in f() have 3 sign changes, so f has 3 or 1 positive real zero(s). f( ) = ( ) ( ) 5 + 3( ) 10( ) 3 ( ) 8( ) 8 = The coefficients in f( ) have 3 sign changes, so f has 3 or 1 negative zero(s). The possible numbers of zeros for f are summarized in the table below. Positive real zeros Negative real zeros Imaginar zeros Total zeros Real-Life Application A tachometer measures the speed (in revolutions per minute, or RPMs) at which an engine shaft rotates. For a certain boat, the speed (in hundreds of RPMs) of the engine shaft and the speed s (in miles per hour) of the boat are modeled b RPM s() = What is the tachometer reading when the boat travels 15 miles per hour? Substitute 15 for s() in the function. You can rewrite the resulting equation as 0 = The related function to this equation is f() = B Descartes s Rule of Signs, ou know f has 3 or 1 positive real zero(s). In the contet of speed, negative real zeros and imaginar zeros do not make sense, so ou do not need to check for them. To approimate the positive real zeros of f, use a graphing calculator. From the graph, there is 1 real zero, The tachometer reading is about 1990 RPMs Zero X= Y=0 0 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Determine the possible numbers of positive real zeros, negative real zeros, and imaginar zeros for the function. 9. f() = f() = WHAT IF? In Eample 5, what is the tachometer reading when the boat travels 0 miles per hour? Section. The Fundamental Theorem of Algebra 01

49 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The epressions 5 + i and 5 i are.. WRITING How man solutions does the polnomial equation ( + 8) 3 ( 1) = 0 have? Eplain. Monitoring Progress and Modeling with Mathematics In Eercises 3 8, identif the number of solutions or zeros. (See Eample 1.) = = Degree: 0. Degree: t 1t 3 + t 1 = 0. f(z) = 7z + z g(s) = s 5 s 3 + s h() = In Eercises 9 1, find all zeros of the polnomial function. (See Eample.) 9. f() = f() = g() = h() = g() = h() = g() = f() = ANALYZING RELATIONSHIPS In Eercises 17 0, determine the number of imaginar zeros for the function with the given degree and graph. Eplain our reasoning. In Eercises 1 8, write a polnomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. (See Eample 3.) 1. 5, 1,., 1, , + i., 5 i 5., 5. 3i, i 7., 1 + i, , + i, ERROR ANALYSIS In Eercises 9 and 30, describe and correct the error in writing a polnomial function with rational coefficients and the given zero(s). 9. Zeros:, 1 + i f () = ( ) [ (1 + i ) ] = ( 1 i ) ( 1 i ) = i + + i = (3 + i ) + ( + i ) 17. Degree: 18. Degree: Zero: + i f () = [ ( + i ) ][ + ( + i ) ] = ( i )( + + i ) = + + i i i i i 0 0 = i 3 0 Chapter Polnomial Functions

50 31. OPEN-ENDED Write a polnomial function of degree with zeros 1,, and i. Justif our answer. 3. REASONING Two zeros of f() = are and. Eplain wh the third zero must also be a real number. In Eercises 33 0, determine the possible numbers of positive real zeros, negative real zeros, and imaginar zeros for the function. (See Eample.) 33. g() =. MODELING WITH MATHEMATICS Over a period of 1 ears, the number N of inland lakes infested with zebra mussels in a certain state can be modeled b N = 0.08t t 3.t t.5 where t is time (in ears). In which ear did the number of infested inland lakes first reach 10? 3. g() = g() = g() = g() = g() = g() = g() = REASONING Which is not a possible classification of zeros for f() = ? Eplain. A B C D three positive real zeros, two negative real zeros, and zero imaginar zeros three positive real zeros, zero negative real zeros, and two imaginar zeros one positive real zero, four negative real zeros, and zero imaginar zeros one positive real zero, two negative real zeros, and two imaginar zeros. USING STRUCTURE Use Descartes s Rule of Signs to determine which function has at least 1 positive real zero. A f() = B f() = C f() = 5 D f() = MODELING WITH MATHEMATICS For the 1 ears that a grocer store has been open, its annual revenue R (in millions of dollars) can be modeled b the function R = ( t + 1t 3 77t + 00t + 13,50) where t is the number of ears since the store opened. In which ear(s) was the revenue $1.5 million?. MAKING AN ARGUMENT Your friend claims that i is a comple zero of the polnomial function f() = i, but that its conjugate is not a zero. You claim that both i and its conjugate must be zeros b the Comple Conjugates Theorem. Who is correct? Justif our answer. 7. MATHEMATICAL CONNECTIONS A solid monument with the dimensions shown is to be built using 1000 cubic feet of marble. What is the value of? 3. MODELING WITH MATHEMATICS From 1890 to 000, the American Indian, Eskimo, and Aleut population P (in thousands) can be modeled b the function P = 0.00t 3 0.t +.9t + 3, where t is the number of ears since In which ear did the population first reach 7,000? (See Eample 5.) 3 ft 3 ft 3 ft 3 ft Section. The Fundamental Theorem of Algebra 03

51 8. THOUGHT PROVOKING Write and graph a polnomial function of degree 5 that has all positive or negative real zeros. Label each -intercept. Then write the function in standard form. 9. WRITING The graph of the constant polnomial function f() = is a line that does not have an -intercepts. Does the function contradict the Fundamental Theorem of Algebra? Eplain. 50. HOW DO YOU SEE IT? The graph represents a polnomial function of degree. = f() a. How man positive real zeros does the function have? negative real zeros? imaginar zeros? b. Use Descartes s Rule of Signs and our answers in part (a) to describe the possible sign changes in the coefficients of f(). 51. FINDING A PATTERN Use a graphing calculator to graph the function f() = ( + 3) n for n =, 3,, 5,, and 7. a. Compare the graphs when n is even and n is odd. b. Describe the behavior of the graph near the zero = 3 as n increases. c. Use our results from parts (a) and (b) to describe the behavior of the graph of g() = ( ) 0 near =. 5. DRAWING CONCLUSIONS Find the zeros of each function. f() = 5 + g() = h() = k() = a. Describe the relationship between the sum of the zeros of a polnomial function and the coefficients of the polnomial function. b. Describe the relationship between the product of the zeros of a polnomial function and the coefficients of the polnomial function. 53. PROBLEM SOLVING You want to save mone so ou can bu a used car in four ears. At the end of each summer, ou deposit $1000 earned from summer jobs into our bank account. The table shows the value of our deposits over the four-ear period. In the table, g is the growth factor 1 + r, where r is the annual interest rate epressed as a decimal. Deposit Year 1 Year Year 3 Year 1st Deposit g 1000g 1000g 3 nd Deposit rd Deposit 1000 th Deposit 1000 a. Cop and complete the table. b. Write a polnomial function that gives the value v of our account at the end of the fourth summer in terms of g. c. You want to bu a car that costs about $300. What growth factor do ou need to obtain this amount? What annual interest rate do ou need? Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Describe the transformation of f() = represented b g. Then graph each function. (Section.1) 5. g() = g() = ( ) + 5. g() = ( 1) 57. g() = 5( + ) Write a function g whose graph represents the indicated transformation of the graph of f. (Sections 1. and.1) 58. f() = ; vertical shrink b a factor of 1 and a reflection in the -ais f() = + 1 3; horizontal stretch b a factor of 9 0. f() = ; reflection in the -ais, followed b a translation units right and 7 units up 0 Chapter Polnomial Functions

52 .7 Transformations of Polnomial Functions Essential Question How can ou transform the graph of a polnomial function? Transforming the Graph of a Cubic Function Work with a partner. The graph of the cubic function f() = 3 f is shown. The graph of each cubic function g represents a transformation of the graph of f. Write a rule for g. Use a graphing calculator to verif our answers. a. g b. g c. g d. g Transforming the Graph of a Quartic Function Work with a partner. The graph of the quartic function f() = f is shown. The graph of each quartic function g represents a transformation of the graph of f. Write a rule for g. Use a graphing calculator to verif our answers. LOOKING FOR STRUCTURE To be proficient in math, ou need to see complicated things, such as some algebraic epressions, as being single objects or as being composed of several objects. a. Communicate Your Answer g 3. How can ou transform the graph of a polnomial function?. Describe the transformation of f() = represented b g() = ( + 1) + 3. Then graph g. b. g Section.7 Transformations of Polnomial Functions 05

53 .7 Lesson What You Will Learn Core Vocabular Previous polnomial function transformations Describe transformations of polnomial functions. Write transformations of polnomial functions. Describing Transformations of Polnomial Functions You can transform graphs of polnomial functions in the same wa ou transformed graphs of linear functions, absolute value functions, and quadratic functions. Eamples of transformations of the graph of f() = are shown below. Core Concept Transformation f() Notation Eamples Horizontal Translation Graph shifts left or right. Vertical Translation Graph shifts up or down. Reflection Graph flips over - or -ais. f( h) f() + k f( ) f() g() = ( 5) g() = ( + ) g() = + 1 g() = g() = ( ) = g() = 5 units right units left 1 unit up units down over -ais over -ais Horizontal Stretch or Shrink Graph stretches awa from or shrinks toward -ais. Vertical Stretch or Shrink Graph stretches awa from or shrinks toward -ais. f(a) a f() g() = () shrink b a factor of 1 g() = ( 1 ) stretch b a factor of g() = 8 stretch b a factor of 8 g() = 1 shrink b a factor of 1 Translating a Polnomial Function Describe the transformation of f() = 3 represented b g() = ( + 5) 3 +. Then graph each function. Notice that the function is of the form g() = ( h) 3 + k. Rewrite the function to identif h and k. g g() = ( ( 5) ) 3 + f h Because h = 5 and k =, the graph of g is a translation 5 units left and units up of the graph of f. k Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Describe the transformation of f() = represented b g() = ( 3) 1. Then graph each function. 0 Chapter Polnomial Functions

54 Transforming Polnomial Functions Describe the transformation of f represented b g. Then graph each function. 1 a. f() =, g() = b. f() = 5, g() = () 5 3 a. Notice that the function is of b. Notice that the function is of the form g() = a, where the form g() = (a) 5 + k, where a = 1. a = and k = 3. So, the graph of g is a So, the graph of g is a reflection in the -ais and a horizontal shrink b a factor of 1 vertical shrink b a factor of and a translation 3 units down 1 of the graph of f. of the graph of f. f g f g Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Describe the transformation of f() = 3 represented b g() = ( + ) 3. Then graph each function. Writing Transformations of Polnomial Functions Writing Transformed Polnomial Functions Let f() = Write a rule for g and then graph each function. Describe the graph of g as a transformation of the graph of f. a. g() = f( ) b. g() = 3f() a. g() = f( ) b. g() = 3f() = ( ) 3 + ( ) + 1 = 3( ) = = REMEMBER Vertical stretches and shrinks do not change the -intercept(s) of a graph. You can observe this using the graph in Eample 3(b). g f 8 g f The graph of g is a reflection The graph of g is a vertical stretch in the -ais of the graph of f. b a factor of 3 of the graph of f. Section.7 Transformations of Polnomial Functions 07

55 Writing a Transformed Polnomial Function Let the graph of g be a vertical stretch b a factor of, followed b a translation 3 units up of the graph of f() =. Write a rule for g. Check 5 3 g h f Step 1 First write a function h that represents the vertical stretch of f. h() = f() Multipl the output b. = ( ) Substitute for f(). = Distributive Propert Step Then write a function g that represents the translation of h. g() = h() + 3 Add 3 to the output. = + 3 Substitute for h(). The transformed function is g() = + 3. Modeling with Mathematics ( 3) ft ft ft The function V() = represents the volume (in cubic feet) of the square pramid shown. The function W() = V(3) represents the volume (in cubic feet) when is measured in ards. Write a rule for W. Find and interpret W(10). 1. Understand the Problem You are given a function V whose inputs are in feet and whose outputs are in cubic feet. You are given another function W whose inputs are in ards and whose outputs are in cubic feet. The horizontal shrink shown b W() = V(3) makes sense because there are 3 feet in 1 ard. You are asked to write a rule for W and interpret the output for a given input.. Make a Plan Write the transformed function W() and then find W(10). 3. Solve the Problem W() = V(3) Net, find W(10). = 1 3 (3)3 (3) Replace with 3 in V(). = Simplif. W(10) = 9(10) 3 9(10) = = 8100 When is 10 ards, the volume of the pramid is 8100 cubic feet.. Look Back Because W(10) = V(30), ou can check that our solution is correct b verifing that V(30) = V(30) = 1 3 (30)3 (30) = = 8100 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 3. Let f() = 5 + and g() = f(). Write a rule for g and then graph each function. Describe the graph of g as a transformation of the graph of f.. Let the graph of g be a horizontal stretch b a factor of, followed b a translation 3 units to the right of the graph of f() = Write a rule for g. 5. WHAT IF? In Eample 5, the height of the pramid is, and the volume (in cubic feet) is represented b V() = 3. Write a rule for W. Find and interpret W(7). 08 Chapter Polnomial Functions

56 .7 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The graph of f() = ( + ) 3 is a translation of the graph of f() = 3.. VOCABULARY Describe how the verte form of quadratic functions is similar to the form f() = a( h) 3 + k for cubic functions. Monitoring Progress and Modeling with Mathematics In Eercises 3, describe the transformation of f represented b g. Then graph each function. (See Eample 1.) 3. f() =, g() = + 3. f() =, g() = ( 5) 5. f() = 5, g() = ( ) 5 1. f() =, g() = ( + 1) ANALYZING RELATIONSHIPS In Eercises 7 10, match the function with the correct transformation of the graph of f. Eplain our reasoning. In Eercises 11 1, describe the transformation of f represented b g. Then graph each function. (See Eample.) 11. f() =, g() = 1. f() =, g() = f() = 3, g() = f() =, g() = f() = 5, g() = 3 ( + )5 1. f() =, g() = () 3 f In Eercises 17 0, write a rule for g and then graph each function. Describe the graph of g as a transformation of the graph of f. (See Eample 3.) 17. f() = + 1, g() = f( + ) 18. f() = 5 + 3, g() = 3f() 7. = f( ) 8. = f( + ) + 9. = f( ) = f() 19. f() = 3 +, g() = f() 0. f() = + 3 1, g() = f( ) 5 1 A. B. 1. ERROR ANALYSIS Describe and correct the error in graphing the function g() = ( + ). C. D. Section.7 Transformations of Polnomial Functions 09

57 . ERROR ANALYSIS Describe and correct the error in describing the transformation of the graph of f() = 5 represented b the graph of g() = (3) 5. The graph of g is a horizontal shrink b a factor of 3, followed b a translation units down of the graph of f. In Eercises 3, write a rule for g that represents the indicated transformations of the graph of f. (See Eample.) 3. f() = 3 ; translation 3 units left, followed b a reflection in the -ais. f() = + + ; vertical stretch b a factor of, followed b a translation units right 5. f() = 3 + 9; horizontal shrink b a factor of 1 3 and a translation units up, followed b a reflection in the -ais 30. THOUGHT PROVOKING Write and graph a transformation of the graph of f() = that results in a graph with a -intercept of. 31. PROBLEM SOLVING A portion of the path that a hummingbird flies while feeding can be modeled b the function 1 f() = 5 ( ) ( 7), 0 7 where is the horizontal distance (in meters) and f() is the height (in meters). The hummingbird feeds each time it is at ground level. a. At what distances does the hummingbird feed? b. A second hummingbird feeds meters farther awa than the first hummingbird and flies twice as high. Write a function to model the path of the second hummingbird.. f() = ; reflection in the -ais and a vertical stretch b a factor of 3, followed b a translation 1 unit down 7. MODELING WITH MATHEMATICS The volume V (in cubic feet) of the pramid is given b V() = 3. The function W() = V(3) gives the volume (in cubic feet) of the pramid when is measured in ards. Write a rule for W. Find and interpret W(5). (See Eample 5.) ft ( ) ft (3 + ) ft 8. MAKING AN ARGUMENT The volume of a cube with side length is given b V() = 3. Your friend claims that when ou divide the volume in half, the volume decreases b a greater amount than when ou divide each side length in half. Is our friend correct? Justif our answer. 9. OPEN-ENDED Describe two transformations of the graph of f() = 5 where the order in which the transformations are performed is important. Then describe two transformations where the order is not important. Eplain our reasoning. Maintaining Mathematical Proficienc 3. HOW DO YOU SEE IT? Determine the real zeros of each function. Then describe the transformation of the graph of f that results in the graph of g. 33. MATHEMATICAL CONNECTIONS Write a function V for the volume (in cubic ards) of the right circular cone shown. Then write a function W that gives the volume (in cubic ards) of the cone when is measured in feet. Find and interpret W(3). Reviewing what ou learned in previous grades and lessons Find the minimum value or maimum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing. (Section.) 3. h() = ( + 5) f() = 3. f() = 3( 10)( + ) 37. g() = ( + )( + 8) 38. h() = 1 ( 1) f() = f g 8 ( + 3) d 3 d 10 Chapter Polnomial Functions

58 .8 Analzing Graphs of Polnomial Functions Essential Question How man turning points can the graph of a polnomial function have? A turning point of the graph of a polnomial function is a point on the graph at which the function changes from increasing to decreasing, or decreasing to increasing. turning point turning point Approimating Turning Points ATTENDING TO PRECISION To be proficient in math, ou need to epress numerical answers with a degree of precision appropriate for the problem contet. Work with a partner. Match each polnomial function with its graph. Eplain our reasoning. Then use a graphing calculator to approimate the coordinates of the turning points of the graph of the function. Round our answers to the nearest hundredth. a. f() = + 3 b. f() = c. f() = d. f() = e. f() = f. f() = A. B. C. D. 3 E. F. 7 Communicate Your Answer. How man turning points can the graph of a polnomial function have? 3. Is it possible to sketch the graph of a cubic polnomial function that has no turning points? Justif our answer. Section.8 Analzing Graphs of Polnomial Functions 11

59 .8 Lesson What You Will Learn Core Vocabular local maimum, p. 1 local minimum, p. 1 even function, p. 15 odd function, p. 15 Previous end behavior increasing decreasing smmetric about the -ais Use -intercepts to graph polnomial functions. Use the Location Principle to identif zeros of polnomial functions. Find turning points and identif local maimums and local minimums of graphs of polnomial functions. Identif even and odd functions. Graphing Polnomial Functions In this chapter, ou have learned that zeros, factors, solutions, and -intercepts are closel related concepts. Here is a summar of these relationships. Concept Summar Zeros, Factors, Solutions, and Intercepts Let f() = a n n + a n 1 n a 1 + a 0 be a polnomial function. The following statements are equivalent. Zero: k is a zero of the polnomial function f. Factor: k is a factor of the polnomial f(). Solution: k is a solution (or root) of the polnomial equation f() = 0. -Intercept: If k is a real number, then k is an -intercept of the graph of the polnomial function f. The graph of f passes through (k, 0). Using -Intercepts to Graph a Polnomial Function Graph the function f() = 1 ( + 3)( ). Step 1 Plot the -intercepts. Because 3 and are zeros of f, plot ( 3, 0) and (, 0). ( 3, 0) Step Plot points between and beond the -intercepts (, 0) Step 3 Determine end behavior. Because f() has three factors of the form k and a constant factor of 1, f is a cubic function with a positive leading coefficient. So, f() as and f() + as +. Step Draw the graph so that it passes through the plotted points and has the appropriate end behavior. Monitoring Progress Graph the function. Help in English and Spanish at BigIdeasMath.com 1. f() = 1 ( + 1)( ). f() = 1 ( + )( 1)( 3) 1 Chapter Polnomial Functions

60 The Location Principle You can use the Location Principle to help ou find real zeros of polnomial functions. Core Concept The Location Principle If f is a polnomial function, and a and b are two real numbers such that f(a) < 0 and f(b) > 0, then f has at least one real zero between a and b. To use this principle to locate real zeros of a polnomial function, find a value a at which the polnomial function is negative and another value b at which the function is positive. You can conclude that the function has at least one real zero between a and b. f(b) f(a) a b zero Check 5 0 Zero X=1.5 Y=0 0 5 Find all real zeros of f() = Locating Real Zeros of a Polnomial Function Step 1 Use a graphing calculator to make a table. Step Use the Location Principle. From the table shown, ou can see that f(1) < 0 and f() > 0. So, b the Location Principle, f has a zero between 1 and. Because f is a polnomial function of degree 3, it has three zeros. The onl possible rational zero between 1 and is 3. Using snthetic division, ou can confirm that 3 is a zero. Step 3 Write f() in factored form. Dividing f() b its known factor 3 gives a quotient of So, ou can factor f() as f() = ( 3 ) ( ) = ( 3 ) ( ) = ( 3 ) (3 + 1)( + ). From the factorization, there are three zeros. The zeros of f are 3, 1 3, and. Check this b graphing f. Monitoring Progress 3. Find all real zeros of f() = X X=1 Y Help in English and Spanish at BigIdeasMath.com Section.8 Analzing Graphs of Polnomial Functions 13

61 READING Local maimum and local minimum are sometimes referred to as relative maimum and relative minimum. Turning Points Another important characteristic of graphs of polnomial functions is that the have turning points corresponding to local maimum and minimum values. The -coordinate of a turning point is a local maimum of the function when the point is higher than all nearb points. The -coordinate of a turning point is a local minimum of the function when the point is lower than all nearb points. The turning points of a graph help determine the intervals for which a function is increasing or decreasing. function is decreasing local maimum function is increasing function is increasing local minimum Core Concept Turning Points of Polnomial Functions 1. The graph of ever polnomial function of degree n has at most n 1 turning points.. If a polnomial function of degree n has n distinct real zeros, then its graph has eactl n 1 turning points. Finding Turning Points Graph each function. Identif the -intercepts and the points where the local maimums and local minimums occur. Determine the intervals for which each function is increasing or decreasing. a. f() = b. g() = a. Use a graphing calculator to graph the function. The graph of f has one -intercept and two turning points. Use the graphing calculator s zero, maimum, and minimum features to approimate the coordinates of the points. 5 The -intercept of the graph is 1.0. The function has a local maimum at (0, ) and a local minimum at (, ). The function is increasing when < 0 and > and decreasing when 0 < <. 3 Maimum X=0 Y= 10 0 b. Use a graphing calculator to graph the function. The graph of g has four -intercepts and three turning points. Use the graphing calculator s zero, maimum, and minimum features to approimate the coordinates of the points. Minimum X= Y= The -intercepts of the graph are 1.1, 0.9, 1.8, and The function has a local maimum at (1.11, 5.11) and local minimums at ( 0.57,.51) and (3.9, 3.0). The function is increasing when 0.57 < < 1.11 and > 3.9 and decreasing when < 0.57 and 1.11 < < 3.9. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Graph f() = Identif the -intercepts and the points where the local maimums and local minimums occur. Determine the intervals for which the function is increasing or decreasing. 1 Chapter Polnomial Functions

62 Even and Odd Functions Core Concept Even and Odd Functions A function f is an even function when f( ) = f() for all in its domain. The graph of an even function is smmetric about the -ais. A function f is an odd function when f( ) = f() for all in its domain. The graph of an odd function is smmetric about the origin. One wa to recognize a graph that is smmetric about the origin is that it looks the same after a 180 rotation about the origin. Even Function Odd Function (, ) (, ) (, ) (, ) For an even function, if (, ) is on the For an odd function, if (, ) is on the graph, then (, ) is also on the graph. graph, then (, ) is also on the graph. Identifing Even and Odd Functions Determine whether each function is even, odd, or neither. a. f() = 3 7 b. g() = + 1 c. h() = 3 + a. Replace with in the equation for f, and then simplif. f( ) = ( ) 3 7( ) = = ( 3 7) = f() Because f( ) = f(), the function is odd. b. Replace with in the equation for g, and then simplif. g( ) = ( ) + ( ) 1 = + 1 = g() Because g( ) = g(), the function is even. c. Replacing with in the equation for h produces h( ) = ( ) 3 + = 3 +. Because h() = 3 + and h() = 3, ou can conclude that h( ) h() and h( ) h(). So, the function is neither even nor odd. Monitoring Progress Determine whether the function is even, odd, or neither. Help in English and Spanish at BigIdeasMath.com 5. f() = + 5. f() = f() = 5 Section.8 Analzing Graphs of Polnomial Functions 15

63 .8 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE A local maimum or local minimum of a polnomial function occurs at a point of the graph of the function.. WRITING Eplain what a local maimum of a function is and how it ma be different from the maimum value of the function. Monitoring Progress and Modeling with Mathematics ANALYZING RELATIONSHIPS In Eercises 3, match the function with its graph. 3. f() = ( 1)( )( + ). h() = ( + ) ( + 1) 5. g() = ( + 1)( 1)( + ). f() = ( 1) ( + ) A. B. ERROR ANALYSIS In Eercises 15 and 1, describe and correct the error in using factors to graph f. 15. f() = ( + )( 1) C D. 1. f() = ( 3) 3 In Eercises 7 1, graph the function. (See Eample 1.) 7. f() = ( ) ( + 1) 8. f() = ( + ) ( + ) 9. h() = ( + 1) ( 1)( 3) 10. g() = ( + 1)( + )( 1) 11. h() = 1 ( 5)( + )( 3) 3 1. g() = 1 ( + )( + 8)( 1) h() = ( 3)( + + 1) In Eercises 17, find all real zeros of the function. (See Eample.) 17. f() = f() = h() = h() = g() = f() = f() = ( )( + 1) 1 Chapter Polnomial Functions

64 In Eercises 3 30, graph the function. Identif the -intercepts and the points where the local maimums and local minimums occur. Determine the intervals for which the function is increasing or decreasing. (See Eample 3.) 3. g() = g() = h() = 3 +. f() = f() = f() = h() = g() = In Eercises 31 3, estimate the coordinates of each turning point. State whether each corresponds to a local maimum or a local minimum. Then estimate the real zeros and find the least possible degree of the function The graph of f has -intercepts at = 3, = 1, and = 5. f has a local maimum value when = 1. f has a local minimum value when = and when =. In Eercises 39, determine whether the function is even, odd, or neither. (See Eample.) 39. h() = 7 0. g() = + 1. f() = + 3. f() = g() = f() = f() = 1. h() = USING TOOLS When a swimmer does the breaststroke, the function S = 1t t 1870t t 737t 3 + 1t.3t models the speed S (in meters per second) of the swimmer during one complete stroke, where t is the number of seconds since the start of the stroke and 0 t 1.. Use a graphing calculator to graph the function. At what time during the stroke is the swimmer traveling the fastest? OPEN-ENDED In Eercises 37 and 38, sketch a graph of a polnomial function f having the given characteristics. 37. The graph of f has -intercepts at =, = 0, and =. f has a local maimum value when = 1. f has a local minimum value when =. 8. USING TOOLS During a recent period of time, the number S (in thousands) of students enrolled in public schools in a certain countr can be modeled b S = ,300, where is time (in ears). Use a graphing calculator to graph the function for the interval 0 1. Then describe how the public school enrollment changes over this period of time. 9. WRITING Wh is the adjective local, used to describe the maimums and minimums of cubic functions, sometimes not required for quadratic functions? Section.8 Analzing Graphs of Polnomial Functions 17

65 50. HOW DO YOU SEE IT? The graph of a polnomial function is shown. = f() a. Find the zeros, local maimum, and local minimum values of the function. b. Compare the -intercepts of the graphs of = f() and = f(). c. Compare the maimum and minimum values of the functions = f() and = f(). 53. PROBLEM SOLVING Quonset huts are temporar, all-purpose structures shaped like half-clinders. You have 1100 square feet of material to build a quonset hut. a. The surface area S of a quonset hut is given b S = πr + πr. Substitute 1100 for S and then write an epression for in terms of r. b. The volume V of a quonset hut is given b V = 1 πr. Write an equation that gives V as a function in terms of r onl. c. Find the value of r that maimizes the volume of the hut. 51. MAKING AN ARGUMENT Your friend claims that the product of two odd functions is an odd function. Is our friend correct? Eplain our reasoning. 5. MODELING WITH MATHEMATICS You are making a rectangular bo out of a 1-inch-b-0-inch piece of cardboard. The bo will be formed b making the cuts shown in the diagram and folding up the sides. You want the bo to have the greatest volume possible. 0 in. 1 in. a. How long should ou make the cuts? b. What is the maimum volume? c. What are the dimensions of the finished bo? 5. THOUGHT PROVOKING Write and graph a polnomial function that has one real zero in each of the intervals < < 1, 0 < < 1, and < < 5. Is there a maimum degree that such a polnomial function can have? Justif our answer. 55. MATHEMATICAL CONNECTIONS A clinder is inscribed in a sphere of radius 8 inches. Write an equation for the volume of the clinder as a function of h. Find the value of h that maimizes the volume of the inscribed clinder. What is the maimum volume of the clinder? h 8 in. Maintaining Mathematical Proficienc State whether the table displas linear data, quadratic data, or neither. Eplain. (Section.) 5. Months, Savings (dollars), Reviewing what ou learned in previous grades and lessons Time (seconds), Height (feet), Chapter Polnomial Functions

66 .9 Modeling with Polnomial Functions Essential Question Essential Question How can ou find a polnomial model for real-life data? Modeling Real-Life Data Work with a partner. The distance a baseball travels after it is hit depends on the angle at which it was hit and the initial speed. The table shows the distances a baseball hit at an angle of 35 travels at various initial speeds. Initial speed, (miles per hour) Distance, (feet) a. Recall that when data have equall-spaced -values, ou can analze patterns in the differences of the -values to determine what tpe of function can be used to model the data. If the first differences are constant, then the set of data fits a linear model. If the second differences are constant, then the set of data fits a quadratic model. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. Find the first and second differences of the data. Are the data linear or quadratic? Eplain our reasoning b. Use a graphing calculator to draw a scatter plot of the data. Do the data appear linear or quadratic? Use the regression feature of the graphing calculator to find a linear or quadratic model that best fits the data c. Use the model ou found in part (b) to find the distance a baseball travels when it is hit at an angle of 35 and travels at an initial speed of 10 miles per hour. d. According to the Baseball Almanac, An drive over 00 feet is noteworth. A blow of 50 feet shows eceptional power, as the majorit of major league plaers are unable to hit a ball that far. Anthing in the 500-foot range is genuinel historic. Estimate the initial speed of a baseball that travels a distance of 500 feet. Communicate Your Answer. How can ou find a polnomial model for real-life data? 3. How well does the model ou found in Eploration 1(b) fit the data? Do ou think the model is valid for an initial speed? Eplain our reasoning. Section.9 Modeling with Polnomial Functions 19

67 .9 Lesson What You Will Learn Core Vocabular finite differences, p. 0 Previous scatter plot Write polnomial functions for sets of points. Write polnomial functions using finite differences. Use technolog to find models for data sets. Writing Polnomial Functions for a Set of Points You know that two points determine a line and three points not on a line determine a parabola. In Eample 1, ou will see that four points not on a line or a parabola determine the graph of a cubic function. Writing a Cubic Function Check Check the end behavior of f. The degree of f is odd and a < 0. So, f() + as and f() as +, which matches the graph. Write the cubic function whose graph is shown. Step 1 Use the three -intercepts to write the function in factored form. f() = a( + )( 1)( 3) Step Find the value of a b substituting the coordinates of the point (0, ). = a(0 + )(0 1)(0 3) = 1a 1 = a The function is f() = ( + )( 1)( 3). 1 (, 0) (3, 0) (1, 0) (0, ) 1 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write a cubic function whose graph passes through the given points. 1. (, 0), (0, 10), (, 0), (5, 0). ( 1, 0), (0, 1), (, 0), (3, 0) Finite Differences When the -values in a data set are equall spaced, the differences of consecutive -values are called finite differences. Recall from Section. that the first and second differences of = are: equall-spaced -values first differences: second differences: Notice that = has degree two and that the second differences are constant and nonzero. This illustrates the first of the two properties of finite differences shown on the net page. 0 Chapter Polnomial Functions

68 Core Concept Properties of Finite Differences 1. If a polnomial function = f() has degree n, then the nth differences of function values for equall-spaced -values are nonzero and constant.. Conversel, if the nth differences of equall-spaced data are nonzero and constant, then the data can be represented b a polnomial function of degree n. The second propert of finite differences allows ou to write a polnomial function that models a set of equall-spaced data. Writing a Function Using Finite Differences Use finite differences to determine the degree of the polnomial function that fits the data. Then use technolog to find the polnomial function f () Step 1 Write the function values. Find the first differences b subtracting consecutive values. Then find the second differences b subtracting consecutive first differences. Continue until ou obtain differences that are nonzero and constant. f (1) f() f(3) f() f(5) f() f(7) Write function values for equall-spaced -values. First differences Second differences Third differences Because the third differences are nonzero and constant, ou can model the data eactl with a cubic function. Step Enter the data into a graphing calculator and use CubicReg cubic regression to obtain a polnomial function. =a 3 +b +c+d 1 Because 0.17, 1 = 0.5, and , a polnomial function that 3 fits the data eactl is f() = a=.17 b=.5 c= d=0 R =1 Monitoring Progress 3. Use finite differences to determine the degree of the polnomial function that fits the data. Then use technolog to find the polnomial function. Help in English and Spanish at BigIdeasMath.com f() Section.9 Modeling with Polnomial Functions 1

69 Finding Models Using Technolog In Eamples 1 and, ou found a cubic model that eactl fits a set of data. In man real-life situations, ou cannot find models to fit data eactl. Despite this limitation, ou can still use technolog to approimate the data with a polnomial model, as shown in the net eample. Real-Life Application The table shows the total U.S. biomass energ consumptions (in trillions of British thermal units, or Btus) in the ear t, where t = 1 corresponds to 001. Find a polnomial model for the data. Use the model to estimate the total U.S. biomass energ consumption in 013. t t According to the U.S. Department of Energ, biomass includes agricultural and forestr residues, municipal solid wastes, industrial wastes, and terrestrial and aquatic crops grown solel for energ purposes. Among the uses for biomass is production of electricit and liquid fuels such as ethanol. Step 1 Enter the data into a graphing calculator and make a scatter plot. The data suggest a cubic model Step 3 Check the model b graphing it and the data in the same viewing window. 500 Step Use the cubic regression feature. The polnomial model is =.55t t 118.1t CubicReg =a 3 +b +c+d a= b= c= d= R = Step Use the trace feature to estimate the value of the model when t = Y1= ^3+_ X=13 Y= The approimate total U.S. biomass energ consumption in 013 was about 385 trillion Btus. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use a graphing calculator to find a polnomial function that fits the data Chapter Polnomial Functions

70 .9 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE When the -values in a set of data are equall spaced, the differences of consecutive -values are called.. WRITING Eplain how ou know when a set of data could be modeled b a cubic function. Monitoring Progress and Modeling with Mathematics In Eercises 3, write a cubic function whose graph is shown. (See Eample 1.) 3.. ( 1, 0) (0, ) (, 0) (1, 0) 5.. ( 5, 0) 8 (1, 0) 8 (, 0) 8 (, ) 8 (, ) 8 (, 0) ( 3, 0) ( 1, 0) ( 3, 0) (, 0) (0, 9) (3, 0) 8 In Eercises 7 1, use finite differences to determine the degree of the polnomial function that fits the data. Then use technolog to find the polnomial function. (See Eample.) f() f() (, 317), ( 3, 37), (, 1), ( 1, 7), (0, 1), (1, 3), (, 7), (3, 89), (, 933) 10. (, 7), (, 15), (, ), (0, ), (, 1), (, 15), (, 8), (8, 07), (10, 98) 11. (, 98), ( 1, ), (0, 1), (1, ), (, ), (3, ), (, ), (5, ), (, 1) 1. (1, 0), (, ), (3, ), (, ), (5, 1), (, 10), (7, 11), (8, 378), (9, 90) 13. ERROR ANALYSIS Describe and correct the error in writing a cubic function whose graph passes through the given points. (, 0), (1, 0), (3, 0), (0, 5) 5 = a(0 )(0 + 1)(0 + 3) 5 = 18a a = 3 f () = 3( )( + 1)( + 3) 1. MODELING WITH MATHEMATICS The dot patterns show pentagonal numbers. The number of dots in the nth pentagonal number is given b f(n) = 1 n(3n 1). Show that this function has constant second-order differences. 15. OPEN-ENDED Write three different cubic functions that pass through the points (3, 0), (, 0), and (, ). Justif our answers. 1. MODELING WITH MATHEMATICS The table shows the ages of cats and their corresponding ages in human ears. Find a polnomial model for the data for the first 8 ears of a cat s life. Use the model to estimate the age (in human ears) of a cat that is 3 ears old. (See Eample 3.) Age of cat, Human ears, Section.9 Modeling with Polnomial Functions 3

71 17. MODELING WITH MATHEMATICS The data in the table show the average speeds (in miles per hour) of a pontoon boat for several different engine speeds (in hundreds of revolutions per minute, or RPMs). Find a polnomial model for the data. Estimate the average speed of the pontoon boat when the engine speed is 800 RPMs HOW DO YOU SEE IT? The graph shows tpical speeds (in feet per second) of a space shuttle seconds after it is launched. Shuttle speed (feet per second) Space Launch Time (seconds) a. What tpe of polnomial function models the data? Eplain. b. Which nth-order finite difference should be constant for the function in part (a)? Eplain. 19. MATHEMATICAL CONNECTIONS The table shows the number of diagonals for polgons diagonal with n sides. Find a polnomial function that fits the data. Determine the total number of diagonals in the decagon shown. Number of sides, n Number of diagonals, d MAKING AN ARGUMENT Your friend states that it is not possible to determine the degree of a function given the first-order differences. Is our friend correct? Eplain our reasoning. 1. WRITING Eplain wh ou cannot alwas use finite differences to find a model for real-life data sets.. THOUGHT PROVOKING A, B, and C are zeros of a cubic polnomial function. Choose values for A, B, and C such that the distance from A to B is less than or equal to the distance from A to C. Then write the function using the A, B, and C values ou chose. 3. MULTIPLE REPRESENTATIONS Order the polnomial functions according to their degree, from least to greatest. A. f() = B. C. D. g h() k() ABSTRACT REASONING Substitute the epressions z, z + 1, z +,, z + 5 for in the function f() = a 3 + b + c + d to generate si equallspaced ordered pairs. Then show that the third-order differences are constant. Maintaining Mathematical Proficienc Solve the equation using square roots. (Section 3.1) 5. = = ( 3) = 8. ( + 3 5) = Solve the equation using the Quadratic Formula. (Section 3.) = = = = Reviewing what ou learned in previous grades and lessons Chapter Polnomial Functions

72 .5.9 What Did You Learn? Core Vocabular repeated solution, p. 190 comple conjugates, p. 199 local maimum, p. 1 local minimum, p. 1 even function, p. 15 odd function, p. 15 finite differences, p. 0 Core Concepts Section.5 The Rational Root Theorem, p. 191 The Irrational Conjugates Theorem, p. 193 Section. The Fundamental Theorem of Algebra, p. 198 The Comple Conjugates Theorem, p. 199 Descartes s Rule of Signs, p. 00 Section.7 Transformations of Polnomial Functions, p. 0 Writing Transformed Polnomial Functions, p. 07 Section.8 Zeros, Factors, Solutions, and Intercepts of Polnomials, p. 1 The Location Principle, p. 13 Turning Points of Polnomial Functions, p. 1 Even and Odd Functions, p. 15 Section.9 Writing Polnomial Functions for Data Sets, p. 0 Properties of Finite Differences, p. 1 Mathematical Practices 1. Eplain how understanding the Comple Conjugates Theorem allows ou to construct our argument in Eercise on page 03.. Describe how ou use structure to accuratel match each graph with its transformation in Eercises 7 10 on page 09. Performance Task For the Birds -- Wildlife Management How does the presence of humans affect the population of sparrows in a park? Do more humans mean fewer sparrows? Or does the presence of humans increase the number of sparrows up to a point? Are there a minimum number of sparrows that can be found in a park, regardless of how man humans there are? What can a mathematical model tell ou? To eplore the answers to these questions and more, go to BigIdeasMath.com. 5

73 Chapter Review.1 Graphing Polnomial Functions (pp ) Dnamic Solutions available at BigIdeasMath.com Graph f() = To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior ( 3, 1) (, 0) (, ) f() The degree is odd and the leading coefficient is positive. So, f() as and f() + as +. 1 (1, 9) (0, 10) Decide whether the function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient. 1. h() = p() = Graph the polnomial function. 3. h() = f() = g() = + +. Adding, Subtracting, and Multipling Polnomials (pp ) a. Multipl ( ), ( 1), and ( + 3) in a horizontal format. ( )( 1)( + 3) = ( 3 + )( + 3) = ( 3 + ) + ( 3 + )3 = = b. Use Pascal s Triangle to epand ( + ). The coefficients from the fourth row of Pascal s Triangle are 1,,,, and 1. ( + ) = 1() + () 3 () + () () + ()() 3 + 1() = Find the sum or difference.. ( 3 1 5) ( ) 7. ( ) + ( 3 + 9) 8. ( ) ( + 1) Find the product. 9. ( + 7)( + 3) 10. (m + n) (s + )(s + )(s 3) Use Pascal s Triangle to epand the binomial. 1. (m + ) 13. (3s + ) 5 1. (z + 1) Chapter Polnomial Functions

74 .3 Dividing Polnomials (pp ) Use snthetic division to evaluate f() = when = The remainder is 7. So, ou can conclude from the Remainder Theorem that f( 3) = 7. You can check this b substituting = 3 in the original function. Check f( 3) = ( 3) 3 + ( 3) + 8( 3) + 10 = = 7 Divide using polnomial long division or snthetic division. 15. ( ) ( + + 1) 1. ( ) ( + + ) 17. ( 7) ( + ) 18. Use snthetic division to evaluate g() = 3 + when = 5.. Factoring Polnomials (pp ) a. Factor + 8 completel. + 8 = ( 3 + 8) Factor common monomial. = ( ) Write as a 3 + b 3. = ( + )( + ) Sum of Two Cubes Pattern b. Determine whether + is a factor of f() = Find f() b snthetic division Because f() = 0, the binomial + is a factor of f() = Factor the polnomial completel z 5 1z z 1. a 3 7a 8a + 8. Show that + is a factor of f() = Then factor f() completel. Chapter Chapter Review 7

75 .5 Solving Polnomial Equations (pp ) a. Find all real solutions of = 0. Step 1 List the possible rational solutions. The leading coefficient of the polnomial f() = is 1, and the constant term is 1. So, the possible rational solutions of f() = 0 are 1 = ± 1, ± 1, ± 3 1, ± 1, ± 1, ± 1 1. Step Test possible solutions using snthetic division until a solution is found Step 3 f() 0, so is not a factor of f(). f( ) = 0, so + is a factor of f(). Factor completel using the result of snthetic division. ( + )( ) = 0 Write as a product of factors. ( + )( + )( 3) = 0 Factor the trinomial. So, the solutions are = and = 3. b. Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the zeros and 1 +. B the Irrational Conjugates Theorem, 1 must also be a zero of f. f() = ( + ) [ ( 1 + ) ][ ( 1 )] Write f() in factored form. = ( + ) [ ( 1) ] [ ( 1) + ] Regroup terms. = ( + ) [ ( 1) ] Multipl. = ( + ) [ ( + 1) ] Epand binomial. = ( + )( 1) Simplif. = Multipl. = Combine like terms. Find all real solutions of the equation = = 0 Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. 5. 1, 3., 3, 5 7., 5, You use 0 cubic inches of cla to make a sculpture shaped as a rectangular prism. The width is inches less than the length and the height is inches more than three times the length. What are the dimensions of the sculpture? Justif our answer. 8 Chapter Polnomial Functions

76 . The Fundamental Theorem of Algebra (pp ) Find all zeros of f() = Step 1 Step Find the rational zeros of f. Because f is a polnomial function of degree, it has four zeros. The possible rational zeros are ±1, ±3, ±9, and ± 7. Using snthetic division, ou can determine that 1 is a zero and 3 is also a zero. Write f() in factored form. Dividing f() b its known factors 1 and + 3 gives a quotient of + 9. So, f() = ( 1)( + 3)( + 9). Step 3 Find the comple zeros of f. Solving + 9 = 0, ou get = ±3i. This means + 9 = ( + 3i)( 3i). f() = ( 1)( + 3)( + 3i)( 3i) From the factorization, there are four zeros. The zeros of f are 1, 3, 3i, and 3i. Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. 9. 3, 1 + i 30. 1,, i 31. 5,, 1 i 3 Determine the possible numbers of positive real zeros, negative real zeros, and imaginar zeros for the function. 3. f() = f() = Transformations of Polnomial Functions (pp ) Describe the transformation of f() = 3 represented b g() = ( ) 3. Then graph each function. Notice that the function is of the form g() = ( h) 3 + k. Rewrite the function to identif h and k. g() = ( ) 3 + ( ) f g h k 8 Because h = and k =, the graph of g is a translation units right and units down of the graph of f. Describe the transformation of f represented b g. Then graph each function. 3. f() = 3, g() = ( ) f() =, g() = ( + 9) Write a rule for g. 3. Let the graph of g be a horizontal stretch b a factor of, followed b a translation 3 units right and 5 units down of the graph of f() = Let the graph of g be a translation 5 units up, followed b a reflection in the -ais of the graph of f() = 3 1. Chapter Chapter Review 9

77 .8 Analzing Graphs of Polnomial Functions (pp ) Graph the function f() = ( + )( ). Then estimate the points where the local maimums and local minimums occur. Step 1 Plot the -intercepts. Because, 0, and are zeros of f, plot (, 0), (0, 0), and (, 0). Step Plot points between and beond the -intercepts. Step 3 Step Determine end behavior. Because f() has three factors of the form k and a constant factor of 1, f is a cubic function with a positive leading coefficient. So f() as and f() + as +. Draw the graph so it passes through the plotted points and has the appropriate end behavior. (, 0) Minimum X=1.15 (0, 0) (, 0) (1, 3) Y= The function has a local maimum at ( 1.15, 3.08) and a local minimum at (1.15, 3.08). Graph the function. Identif the -intercepts and the points where the local maimums and local minimums occur. Determine the intervals for which the function is increasing or decreasing. 38. f() = f() = Determine whether the function is even, odd, or neither. 0. f() = g() = 3 7. h() = Modeling with Polnomial Functions (pp. 19 ) Write the cubic function whose graph is shown. Step 1 Use the three -intercepts to write the function in factored form. f() = a( + 3)( + 1)( ) ( 1, 0) ( 3, 0) 8 (, 0) Step Find the value of a b substituting the coordinates of the point (0, 1). 1 = a(0 + 3)(0 + 1)(0 ) 1 = a = a 1 1 (0, 1) The function is f() = ( + 3)( + 1)( ). 3. Write a cubic function whose graph passes through the points (, 0), (, 0), (0, ), and (, 0).. Use finite differences to determine the degree of the polnomial function that fits the data. Then use technolog to find the polnomial function f() Chapter Polnomial Functions

78 Chapter Test Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. 1. 3, 1.,, 3i Find the product or quotient. 3. ( )( 7 + 5). (3 3 1) ( + 1) 5. ( ) ( + ). ( + 3) 3 7. The graphs of f() = and g() = ( 3) are shown. a. How man zeros does each function have? Eplain. b. Describe the transformation of f represented b g. c. Determine the intervals for which the function g is increasing or decreasing. f g 8. The volume V (in cubic feet) of an aquarium is modeled b the polnomial function V() = , where is the length of the tank. a. Eplain how ou know = is not a possible rational zero. b. Show that 1 is a factor of V(). Then factor V() completel. c. Find the dimensions of the aquarium shown. 9. One special product pattern is (a b) = a ab + b. Using Pascal s Triangle to epand (a b) gives 1a + a( b) + 1( b). Are the two epressions equivalent? Eplain. Volume = 3 ft Can ou use the snthetic division procedure that ou learned in this chapter to divide an two polnomials? Eplain. 11. Let T be the number (in thousands) of new truck sales. Let C be the number (in thousands) of new car sales. During a 10-ear period, T and C can be modeled b the following equations where t is time (in ears). T = 3t 330t t 7500t C = 1t 330t t 5900t a. Find a new model S for the total number of new vehicle sales. b. Is the function S even, odd, or neither? Eplain our reasoning. 1. Your friend has started a golf cadd business. The table shows the profits p (in dollars) of the business in the first 5 months. Use finite differences to find a polnomial model for the data. Then use the model to predict the profit after 7 months. Month, t Profit, p 5 Chapter Chapter Test 31

BIG IDEAS MATH. Ron Larson Laurie Boswell. Sampler

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