Evaluate and Graph Polynomial Functions

Transcription

1 5.2 Evaluate and Graph Polynomial Functions Before You evaluated and graphed linear and quadratic functions. Now You will evaluate and graph other polynomial functions. Why? So you can model skateboarding participation, as in E. 55. Key Vocabulary polynomial polynomial function synthetic substitution end behavior Recall that a monomial is a number, a variable, or a product of numbers and variables. A polynomial is a monomial or a sum of monomials. A polynomial function is a function of the form f() 5 a n n 1 a n 2 1 n a 1 1 a 0 where a n Þ 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 polynomial function is in standard form if its terms are written in descending order of eponents from left to right. Common Polynomial Functions Degree Type Standard form Eample 0 Constant f() 5 a 0 f() Linear f() 5 a 1 1 a 0 f() Quadratic f() 5 a a 1 1 a 0 f() Cubic f() 5 a a a 1 1 a 0 f() Quartic f() 5 a a a a 1 1 a 0 f() E XAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. a. h() } b. g() Ï } 3 1 π 2 c. f() d. k() Solution a. The function is a polynomial function that is already written in standard form. It has degree 4 (quartic) and a leading coefficient of 1. b. The function is a polynomial function written as g() 5 π Ï } 3 in standard form. It has degree 2 (quadratic) and a leading coefficient of π. c. The function is not a polynomial function because the term 3 21 has an eponent that is not a whole number. d. The function is not a polynomial function because the term 2 does not have a variable base and an eponent that is a whole number. 5.2 Evaluate and Graph Polynomial Functions 337

2 E XAMPLE 2 Evaluate by direct substitution Use direct substitution to evaluate f() when 5 3. f() Write original function. f(3) 5 2(3) 4 2 5(3) 3 2 4(3) 1 8 Substitute 3 for Evaluate powers and multiply Simplify. GUIDED PRACTICE for Eamples 1 and 2 Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. 1. f() p() h() π 2 3 Use direct substitution to evaluate the polynomial function for the given value of. 4. f() ; g() ; 5 4 SYNTHETIC SUBSTITUTION Another way to evaluate a polynomial function is to use synthetic substitution. This method, shown in the net eample, involves fewer operations than direct substitution. E XAMPLE 3 Evaluate by synthetic substitution Use synthetic substitution to evaluate f() from Eample 2 when 5 3. AVOID ERRORS The row of coefficients for f() must include a coefficient of 0 for the missing 2 -term. Solution STEP 1 Write the coefficients of f() in order of descending eponents. Write the value at which f() is being evaluated to the left. -value coefficients STEP 2 Bring down the leading coefficient. Multiply the leading coefficient by the -value. Write the product under the second coefficient. Add STEP Multiply the previous sum by the -value. Write the product under the third coefficient. Add. Repeat for all of the remaining coefficients. The final sum is the value of f() at the given -value c Synthetic substitution gives f(3) 5 23, which matches the result in Eample Chapter 5 Polynomials and Polynomial Functions

3 END BEHAVIOR The end behavior of a function s graph is the behavior of the graph as approaches positive infinity (1`) or negative infinity (2`). For the graph of a polynomial function, the end behavior is determined by the function s degree and the sign of its leading coefficient. KEY CONCEPT For Your Notebook End Behavior of Polynomial Functions READING The epression 1` is read as approaches positive infinity. Degree: odd Leading coefficient: positive f() 2` as 2` y f() 1` as 1` Degree: odd Leading coefficient: negative f() 1` as 2` y f() 2` as 1` Degree: even Leading coefficient: positive Degree: even Leading coefficient: negative f() 1` as 2` y f() 1` as 1` y f() 2` as 2` f() 2` as 1` E XAMPLE 4 Standardized Test Practice What is true about the degree and leading coefficient of the polynomial function whose graph is shown? A Degree is odd; leading coefficient is positive 4 y B Degree is odd; leading coefficient is negative C Degree is even; leading coefficient is positive 3 D Degree is even; leading coefficient is negative From the graph, f() 2` as 2` and f() 2` as 1`. So, the degree is even and the leading coefficient is negative. c The correct answer is D. A B C D GUIDED PRACTICE for Eamples 3 and 4 Use synthetic substitution to evaluate the polynomial function for the given value of. 6. f() ; y 7. g() ; Describe the degree and leading coefficient of the polynomial function whose graph is shown Evaluate and Graph Polynomial Functions 339

4 GRAPHING POLYNOMIAL FUNCTIONS To graph a polynomial function, first plot points to determine the shape of the graph s middle portion. Then use what you know about end behavior to sketch the ends of the graph. E XAMPLE 5 Graph polynomial functions Graph (a) f() and (b) f() Solution 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. (22, 3) 1 y (1, 0) y (2, 21) The degree is odd and leading coefficient is negative. So, f() 1` as 2` and f() 2` as 1`. (21, 24) (0, 23) 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 y (0, 4) (21, 2) 1 (1, 0) 3 y The degree is even and leading coefficient is positive. So, f() 1` as 2` and f() 1` as 1`. (2, 24) at classzone.com E XAMPLE 6 Solve a multi-step problem PHYSICAL SCIENCE The energy E (in foot-pounds) in each square foot of a wave is given by the model E s 4 where s is the wind speed (in knots). Graph the model. Use the graph to estimate the wind speed needed to generate a wave with 1000 foot-pounds of energy per square foot. Solution STEP 1 STEP 2 Make a table of values. The model only deals with positive values of s. s E Plot the points and connect them with a smooth curve. Because the leading coefficient is positive and the degree is even, the graph rises to the right. Energy per square foot (foot-pounds) STEP 3 Eamine the graph to see that s < 24 when E c The wind speed needed to generate the wave is about 24 knots. E Wave Energy (24, 1000) s Wind speed (knots) 340 Chapter 5 Polynomials and Polynomial Functions

5 GUIDED PRACTICE for Eamples 5 and 6 Graph the polynomial function. 9. f() f() f() WHAT IF? If wind speed is measured in miles per hour, the model in Eample 6 becomes E s 4. Graph this model. What wind speed is needed to generate a wave with 2000 foot-pounds of energy per square foot? 5.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 21, 27, and 57 5 STANDARDIZED TEST PRACTICE Es. 2, 24, 37, 50, 52, and 59 5 MULTIPLE REPRESENTATIONS E VOCABULARY Identify the degree, type, leading coefficient, and constant term of the polynomial function f() WRITING Eplain what is meant by the end behavior of a polynomial function. EXAMPLE 1 on p. 337 for Es. 3 8 POLYNOMIAL FUNCTIONS Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. 3. f() f() g() 5 π 4 1 Ï } 6 6. h() 5 3 Ï } h() 5 2 } g() } 2 EXAMPLE 2 on p. 338 for Es DIRECT SUBSTITUTION Use direct substitution to evaluate the polynomial function for the given value of. 9. f() ; f() ; g() ; h() ; h() } } ; g() ; 5 22 EXAMPLE 3 on p. 338 for Es SYNTHETIC SUBSTITUTION Use synthetic substitution to evaluate the polynomial function for the given value of. 15. f() ; f() ; g() ; h() ; f() ; g() ; h() ; f() ; ERROR ANALYSIS Describe and correct the error in evaluating the polynomial function f() when Evaluate and Graph Polynomial Functions 341

6 EXAMPLE 4 on p. 339 for Es MULTIPLE CHOICE The graph of a polynomial function is shown. What is true about the function s degree and leading coefficient? A The degree is odd and the leading coefficient is positive. B The degree is odd and the leading coefficient is negative. 2 y 1 C The degree is even and the leading coefficient is positive. D The degree is even and the leading coefficient is negative. USING END BEHAVIOR Describe the degree and leading coefficient of the polynomial function whose graph is shown. 25. y y 27. y DESCRIBING END BEHAVIOR Describe the end behavior of the graph of the polynomial function by completing these statements: f()? as 2` and f()? as 1`. 28. f() f() f() f() f() f() f() f() f() OPEN-ENDED MATH Write a polynomial function f of degree 5 such that the end behavior of the graph of f is given by f() 1` as 2` and f() 2` as 1`. Then graph the function to verify your answer. EXAMPLE 5 on p. 340 for Es GRAPHING POLYNOMIALS Graph the polynomial function. 38. f() f() f() f() f() f() f() f() f() f() f() f() MULTIPLE CHOICE Which function is represented by the graph shown? 2 y A f() 5 1 } B f() } C f() 5 1 } D f() } VISUAL THINKING Suppose f() 1` as 2` and f() 2` as 1`. Describe the end behavior of g() 5 2f(). 52. SHORT RESPONSE A cubic polynomial function f has leading coefficient 2 and constant term 25. If f(1) 5 0 and f(2) 5 3, what is f(25)? Eplain how you found your answer. 5 WORKED-OUT SOLUTIONS 342 Chapter 5 Polynomials on p. WS1 and Polynomial Functions 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

7 53. CHALLENGE Let f() 5 3 and g() a. Copy and complete the table. b. Use the numbers in the table to complete this statement: As 1`, } f() g()?. c. Eplain how the result from part (b) shows that the functions f and g have the same end behavior as 1`. f () g() f() } g() 10??? 20??? 50??? 100??? 200??? PROBLEM SOLVING EXAMPLE 6 on p. 340 for Es DIAMONDS The weight of an ideal round-cut diamond can be modeled by w d d d where w is the diamond s weight (in carats) and d is its diameter (in millimeters). According to the model, what is the weight of a diamond with a diameter of 15 millimeters? 55. SKATEBOARDING From 1992 to 2003, the number of people in the United States who participated in skateboarding can be modeled by S t t t t where S is the number of participants (in millions) and t is the number of years since Graph the model. Then use the graph to estimate the first year that the number of skateboarding participants was greater than 8 million. 56. MULTIPLE REPRESENTATIONS From 1987 to 2003, the number of indoor movie screens M in the United States can be modeled by M t t t 1 21,600 where t is the number of years since a. Classifying a Function State the degree and type of the function. b. Making a Table Make a table of values for the function. c. Sketching a Graph Use your table to graph the function. 57. SNOWBOARDING From 1992 to 2003, the number of people in the United States who participated in snowboarding can be modeled by S t t t t where S is the number of participants (in millions) and t is the number of years since Graph the model. Use the graph to estimate the first year that the number of snowboarding participants was greater than 2 million. 5.2 Evaluate and Graph Polynomial Functions 343

8 58. MULTI-STEP PROBLEM From 1980 to 2002, the number of quarterly periodicals P published in the United States can be modeled by P t t t t where t is the number of years since a. Describe the end behavior of the graph of the model. b. Graph the model on the domain 0 t 22. c. Use the model to predict the number of quarterly periodicals in the year Is it appropriate to use the model to make this prediction? Eplain. 59. EXTENDED RESPONSE The weight of Sarus crane chicks S and hooded crane chicks H (both in grams) during the 10 days following hatching can be modeled by the functions S t t t H t t t where t is the number of days after hatching. a. Calculate According to the models, what is the difference in weight between 5-day-old Sarus crane chicks and hooded crane chicks? b. Graph Sketch the graphs of the two models. c. Apply A biologist finds that the weight of a crane chick after 3 days is 130 grams. What species of crane is the chick more likely to be? Eplain how you found your answer. 60. CHALLENGE The weight y (in pounds) of a rainbow trout can be modeled by y where is the length of the trout (in inches). a. Write a function that relates the weight y and length of a rainbow trout if y is measured in kilograms and is measured in centimeters. Use the fact that 1 kilogram ø 2.20 pounds and 1 centimeter ø inch. b. Graph the original function and the function from part (a) in the same coordinate plane. What type of transformation can you apply to the graph of y to produce the graph from part (a)? MIXED REVIEW Solve the equation or inequality b b (p. 18) n (p. 18) < 6y 2 1 < 5 (p. 41) (p. 252) q q 5 21 (p. 259) 66. z 2 1 5z < 36 (p. 300) The variables and y vary directly. Write an equation that relates and y. Then find the value of when y (p. 107) , y , y , y , y , y , y PREVIEW Prepare for Lesson 5.3 in Es Write the quadratic function in standard form. (p. 245) 73. y 5 ( 1 3)( 2 7) 74. y 5 8( 2 4)( 1 2) 75. y 5 23( 2 5) y 5 2.5( 2 6) y 5 } 1 2 ( 2 4)2 78. y 5 2 } 5 ( 1 4)( 1 9) Chapter 5 EXTRA Polynomials PRACTICE and Polynomial for Lesson Functions 5.2, p ONLINE QUIZ at classzone.com

9 Use after Lesson Set a Good Viewing Window classzone.com Keystrokes QUESTION What is a good viewing window for a polynomial function? When you graph a function with a graphing calculator, you should choose a viewing window that displays the important characteristics of the graph. EXAMPLE Graph a polynomial function Graph f() STEP 1 Graph the function Graph the function in the standard viewing window. STEP 2 Adjust horizontally Adjust the horizontal scale so that the end behavior of the graph as 1` is visible. STEP 3 Adjust vertically Adjust the vertical scale so that the turning points and end behavior of the graph as 2` are visible , 210 y , 210 y , 220 y 10 P RACTICE Find intervals for and y that describe a good viewing window for the graph of the polynomial function. 1. f() f() f() f() f() f() f() f() REASONING Let g() 5 f() 1 c where f() and g() are polynomial functions and c is a positive constant. How is a good viewing window for the graph of f() related to a good viewing window for the graph of g()? 10. BASEBALL From 1994 to 2003, the average salary S (in thousands of dollars) for major league baseball players can be modeled by S() where is the number of years since Find intervals for the horizontal and vertical aes that describe a good viewing window for the graph of S. 5.2 Evaluate and Graph Polynomial Functions 345

10 5.3 Add, Subtract, and Multiply Polynomials Before You evaluated and graphed polynomial functions. Now You will add, subtract, and multiply polynomials. Why? So you can model collegiate sports participation, as in E. 63. Key Vocabulary like terms, p. 12 To add or subtract polynomials, add or subtract the coefficients of like terms. You can use a vertical or horizontal format. E XAMPLE 1 Add polynomials vertically and horizontally a. Add and in a vertical format. b. Add 3y 3 2 2y 2 2 7y and 24y 2 1 2y 2 5 in a horizontal format. REVIEW SIMPLIFYING For help with simplifying epressions, see p. 10. Solution a b. (3y 3 2 2y 2 2 7y) 1 (24y 2 1 2y 2 5) 5 3y 3 2 2y 2 2 4y 2 2 7y 1 2y y 3 2 6y 2 2 5y 2 5 E XAMPLE 2 Subtract polynomials vertically and horizontally a. Subtract from in a vertical format. b. Subtract 5z 2 2 z 1 3 from 4z 2 1 9z 2 12 in a horizontal format. Solution a. Align like terms, then add the opposite of the subtracted polynomial ( ) b. Write the opposite of the subtracted polynomial, then add like terms. (4z 2 1 9z 2 12) 2 (5z 2 2 z 1 3) 5 4z 2 1 9z z 2 1 z z 2 2 5z 2 1 9z 1 z z z 2 15 GUIDED PRACTICE for Eamples 1 and 2 Find the sum or difference. 1. (t 2 2 6t 1 2) 1 (5t 2 2 t 2 8) 2. (8d d 3 ) 2 (d d 2 2 4) 346 Chapter 5 Polynomials and Polynomial Functions

11 MULTIPLYING POLYNOMIALS To multiply two polynomials, you multiply each term of the first polynomial by each term of the second polynomial. E XAMPLE 3 Multiply polynomials vertically and horizontally a. Multiply 22y 2 1 3y 2 6 and y 2 2 in a vertical format. b. Multiply 1 3 and in a horizontal format. Solution a. 22y 2 1 3y y 2 2 4y 2 2 6y 1 12 Multiply 22y 2 1 3y 2 6 by y 3 1 3y 2 2 6y Multiply 22y 2 1 3y 2 6 by y. 22y 3 1 7y y 1 12 Combine like terms. b. ( 1 3)( ) 5 ( 1 3)3 2 2 ( 1 3)2 1 ( 1 3) E XAMPLE 4 Multiply three binomials Multiply 2 5, 1 1, and 1 3 in a horizontal format. ( 2 5)( 1 1)( 1 3) 5 ( )( 1 3) 5 ( ) 1 ( ) PRODUCT PATTERNS Some binomial products occur so frequently that it is worth memorizing their patterns. You can verify these product patterns by multiplying. KEY CONCEPT For Your Notebook Special Product Patterns Sum and Difference Eample (a 1 b)(a 2 b) 5 a 2 2 b 2 ( 1 4)( 2 4) AVOID ERRORS In general, (a 6 b) 2 Þ a 2 6 b 2 and (a 6 b) 3 Þ a 3 6 b 3. Square of a Binomial Eample (a 1 b) 2 5 a 2 1 2ab 1 b 2 (a 2 b) 2 5 a 2 2 2ab 1 b 2 (y 1 3) 5 y 2 1 6y 1 9 (3z 2 2 5) 2 5 9z z Cube of a Binomial Eample (a 1 b) 3 5 a 3 1 3a 2 b 1 3ab 2 1 b 3 (a 2 b) 3 5 a 3 2 3a 2 b 1 3ab 2 2 b 3 ( 1 2) (p 2 3) 3 5 p 3 2 9p p Add, Subtract, and Multiply Polynomials 347

12 E XAMPLE 5 Use special product patterns a. (3t 1 4)(3t 2 4) 5 (3t) Sum and difference 5 9t b. (8 2 3) 2 5 (8) 2 2 2(8)(3) Square of a binomial c. (pq 1 5) 3 5 (pq) 3 1 3(pq) 2 (5) 1 3(pq)(5) Cube of a binomial 5 p 3 q p 2 q pq GUIDED PRACTICE for Eamples 3, 4, and 5 Find the product. 3. ( 1 2)( ) 4. (a 2 5)(a 1 2)(a 1 6) 5. (y 2 4) 3 E XAMPLE 6 Use polynomial models PETROLEUM Since 1980, the number W (in thousands) of United States wells producing crude oil and the average daily oil output per well O (in barrels) can be modeled by W t t and O t where t is the number of years since Write a model for the average total amount T of crude oil produced per day. What was the average total amount of crude oil produced per day in 2000? DETERMINE SIGNIFICANT DIGITS When multiplying models, round your result so that its terms have the same number of significant digits as the model with the fewest number of significant digits. Solution To find a model for T, multiply the two given models t t t t t t t t t t t Oil refinery in Long Beach, California c Total daily oil output can be modeled by T t t t where T is measured in thousands of barrels. By substituting t 5 20 into the model, you can estimate that the average total amount of crude oil produced per day in 2000 was about 5570 thousand barrels, or 5,570,000 barrels. GUIDED PRACTICE for Eample 6 6. INDUSTRY The models below give the average depth D (in feet) of new wells drilled and the average cost per foot C (in dollars) of drilling a new well. In both models, t represents the number of years since Write a model for the average total cost T of drilling a new well. D 5 109t and C t t Chapter 5 Polynomials and Polynomial Functions

13 5.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 11, 21, and 61 5 STANDARDIZED TEST PRACTICE Es. 2, 15, 47, 56, and VOCABULARY When you add or subtract polynomials, you add or subtract the coefficients of?. 2. WRITING Eplain how a polynomial subtraction problem is equivalent to a polynomial addition problem. EXAMPLES 1 and 2 on p. 346 for Es ADDING AND SUBTRACTING POLYNOMIALS Find the sum or difference. 3. ( ) 1 ( ) 4. ( ) 2 ( ) 5. (4y 2 1 9y 2 5) 2 (4y 2 2 5y 1 3) 6. (z 2 1 5z 2 7) 1 (5z z 2 6) 7. (3s 3 1 s) 1 (4s 3 2 2s 2 1 7s 1 10) 8. (2a 2 2 8) 2 (a 3 1 4a a 1 4) 9. (5c 2 1 7c 1 1) 1 (2c 3 2 6c 1 8) 10. (4t t 2 1 4t) 2 (27t 2 2 5t 1 8) 11. (5b 2 6b 3 1 2b 4 ) 2 (9b 3 1 4b 4 2 7) 12. (3y 2 2 6y y) 1 (5y 4 2 6y 3 1 4y) 13. ( ) 1 ( ) 14. (8v 4 2 2v 2 1 v 2 4) 2 (3v v 2 1 8v) 15. MULTIPLE CHOICE What is the result when is subtracted from ? A B C D EXAMPLE 3 on p. 347 for Es MULTIPLYING POLYNOMIALS Find the product of the polynomials. 16. ( ) (6 1 2) 18. (y 2 7)(y 1 6) 19. (3z 1 1)(z 2 3) 20. (w 1 4)(w 2 1 6w 2 11) 21. (2a 2 3)(a a 2 2) 22. (5c 2 2 4)(2c 2 1 c 2 3) 23. ( )( ) 24. (2d 2 1 4d 1 3)(3d 2 2 7d 1 6) 25. (3y 2 1 6y 2 1)(4y y 2 5) ERROR ANALYSIS Describe and correct the error in simplifying the epression. 26. ( ) 2 ( ) 27. (2 2 7) 3 5 (2) EXAMPLE 4 on p. 347 for Es MULTIPLYING THREE BINOMIALS Find the product of the binomials. 28. ( 1 4)( 2 6)( 2 5) 29. ( 1 1)( 2 7)( 1 3) 30. (z 2 4)(2z 1 2)(z 1 8) 31. (a 2 6)(2a 1 5)(a 1 1) 32. (3p 1 1)(p 1 3)(p 1 1) 33. (b 2 2)(2b 2 1)(2b 1 1) 34. (2s 1 1)(3s 2 2)(4s 2 3) 35. (w 2 6)(4w 2 1)(23w 1 5) 36. (4 2 1)(22 2 7)(25 2 4) 37. (3q 2 8)(29q 1 2)(q 2 2) 5.3 Add, Subtract, and Multiply Polynomials 349

14 EXAMPLE 5 on p. 348 for Es SPECIAL PRODUCTS Find the product. 38. ( 1 5)( 2 5) 39. (w 2 9) (y 1 4) (2c 1 5) (3t 2 4) (5p 2 3)(5p 1 3) 44. (7 2 y) (2a 1 9b)(2a 2 9b) 46. (3z 1 7y) MULTIPLE CHOICE Which epression is equivalent to (3 2 2y) 2? A y 2 B y 2 C y 1 4y 2 D y 1 4y 2 GEOMETRY Write the figure s volume as a polynomial in standard form. 48. V 5 lwh V 5 πr 2 h V 5 s V 5 1 } 3 Bh SPECIAL PRODUCTS Verify the special product pattern by multiplying. 52. (a 1 b)(a 2 b) 5 a 2 2 b (a 1 b) 2 5 a 2 1 2ab 1 b (a 1 b) 3 5 a 3 1 3a 2 b 1 3ab 2 1 b (a 2 b) 3 5 a 3 2 3a 2 b 1 3ab 2 2 b EXTENDED RESPONSE Let p() and q() a. What is the degree of the polynomial p() 1 q()? b. What is the degree of the polynomial p() 2 q()? c. What is the degree of the polynomial p() p q()? d. In general, if p() and q() are polynomials such that p() has degree m, q() has degree n, and m > n, what are the degrees of p() 1 q(), p() 2 q(), and p() p q()? 57. FINDING A PATTERN Look at the following polynomial factorizations ( 2 1)( 1 1) ( 2 1)( ) ( 2 1)( ) a. Factor and into the product of 2 1 and another polynomial. Check your answers by multiplying. b. In general, how can n 2 1 be factored? Show that this factorization works by multiplying the factors. 58. CHALLENGE Suppose f() 5 ( 1 a)( 1 b)( 1 c)( 1 d). If 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. 5 WORKED-OUT SOLUTIONS 350 Chapter 5 Polynomials on p. WS1 and Polynomial Functions 5 STANDARDIZED TEST PRACTICE

15 PROBLEM SOLVING EXAMPLE 6 on p. 348 for Es HIGHER EDUCATION Since 1970, the number (in thousands) of males M and females F attending institutes of higher education can be modeled by M t t t and F t t t where t is the number of years since Write a model for the total number of people attending institutes of higher education. 60. ELECTRONICS From 1999 to 2004, the number of DVD players D (in millions) sold in the United States and the average price per DVD player P (in dollars) can be modeled by D t and P t t where t is the number of years since Write a model for the total revenue R from DVD sales. According to the model, what was the total revenue in 2002? 61. BICYCLING The equation P sF gives the power P (in horsepower) needed to keep a certain bicycle moving at speed s (in miles per hour), where F is the force (in pounds) of road and air resistance. On level ground, the equation F s models the force F. Write a model (in terms of s only) for the power needed to keep the bicycle moving at speed s on level ground. How much power is needed to keep the bicycle moving at 10 miles per hour? at classzone.com 62. MULTI-STEP PROBLEM A dessert is made by taking a hemispherical mound of marshmallow on a 0.5 centimeter thick cookie and covering it with a chocolate shell 1 centimeter thick. Use the diagrams to write two polynomial functions in standard form: M(r) for the combined volume of the marshmallow plus cookie, and D(r) for the volume of the entire dessert. Then use M(r) and D(r) to write a function C(r) for the volume of the chocolate. 63. SHORT RESPONSE From 1997 to 2002, the number of NCAA lacrosse teams for men L m and women L w, as well as the average size of a men s team S m and a women s team S w, can be modeled by L m t and S m t t t L w t and S w t t t where t is the number of years since Write a model for the total number of people N on NCAA lacrosse teams. Eplain how you obtained your model. 5.3 Add, Subtract, and Multiply Polynomials 351

16 64. CHALLENGE From 1970 to 2002, the circulation C (in millions) of Sunday newspapers in the United States can be modeled by C t t t where t is the number of years since Rewrite C as a function of s, where s is the number of years since MIXED REVIEW PREVIEW Prepare for Lesson 5.4 in Es Solve the equation (p. 18) (p. 18) 67. 4t t (p. 18) 68. y 2 2 2y (p. 252) 69. w w (p. 252) (p. 252) 71. 4z z (p. 259) 72. 9a a (p. 259) Solve the system of equations. (p. 178) y 2 2z y 1 z y 2 2z y 1 z y 1 z y 2 z y 1 3z y 1 4z y 1 3z 5 25 Evaluate the determinant of the matri. (p. 203) 76. F G 77. F G 78. 1G F F G QUIZ for Lessons Evaluate the epression. (p. 330) p (2 4 ) 2 3. } } Simplify the epression. (p. 330) 5. ( 4 y 22 )( 23 y 8 ) 6. (a 2 b 25 ) Graph the polynomial function. (p. 337) 3 y 7 } 24 y 0 8. c 3 d 22 } c 5 d g() h() f() Perform the indicated operation. (p. 346) 12. ( ) 2 ( ) 13. ( ) 1 ( ) 14. ( 2 5)( ) 15. ( 1 3)( 2 6)(3 2 1) 16. NATIONAL DEBT On July 21, 2004, the national debt of the United States was about $7,282,000,000,000. The population of the United States at that time was about 294,000,000. Suppose the national debt was divided evenly among everyone in the United States. How much would each person owe? (p. 330) 352 Chapter 5 EXTRA Polynomials PRACTICE and Polynomial for Lesson Functions 5.3, p ONLINE QUIZ at classzone.com

17 5.4 Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find dimensions of archaeological ruins, as in E. 58. Key Vocabulary factored completely factor by grouping quadratic form In Chapter 4, you learned how to factor the following types of quadratic epressions. Type Eample General trinomial (2 1 5)( 2 4) Perfect square trinomial ( 1 4) 2 Difference of two squares (3 1 1)(3 2 1) Common monomial factor (2 1 5) You can also factor polynomials with degree greater than 2. Some of these polynomials can be factored completely using techniques learned in Chapter 4. KEY CONCEPT For Your Notebook Factoring Polynomials Definition A factorable polynomial with integer coefficients is factored completely if it is written as a product of unfactorable polynomials with integer coefficients. Eamples 2( 1 1)( 2 4) and 5 2 ( 2 2 3) are factored completely. 3( 2 2 4) is not factored completely because can be factored as ( 1 2)( 2 2). E XAMPLE 1 Find a common monomial factor Factor the polynomial completely. a ( ) Factor common monomial. 5 ( 1 5)( 2 3) Factor trinomial. b. 2y y 3 5 2y 3 (y 2 2 9) Factor common monomial. 5 2y 3 (y 1 3)(y 2 3) Difference of two squares c. 4z z z 2 5 4z 2 (z 2 2 4z 1 4) Factor common monomial. 5 4z 2 (z 2 2) 2 Perfect square trinomial 5.4 Factor and Solve Polynomial Equations 353

18 FACTORING PATTERNS In part (b) of Eample 1, the special factoring pattern for the difference of two squares is used to factor the epression completely. There are also factoring patterns that you can use to factor the sum or difference of two cubes. KEY CONCEPT For Your Notebook Special Factoring Patterns Sum of Two Cubes Eample a 3 1 b 3 5 (a 1 b)(a 2 2 ab 1 b 2 ) (2) (2 1 3)( ) Difference of Two Cubes Eample a 3 2 b 3 5 (a 2 b)(a 2 1 ab 1 b 2 ) (4) (4 2 1)( ) E XAMPLE 2 Factor the sum or difference of two cubes Factor the polynomial completely. a Sum of two cubes 5 ( 1 4)( ) b. 16z z 2 5 2z 2 (8z ) Factor common monomial. 5 2z 2 F (2z) G Difference of two cubes 5 2z 2 (2z 2 5)(4z z 1 25) GUIDED PRACTICE for Eamples 1 and 2 Factor the polynomial completely y y b b 2 4. w FACTORING BY GROUPING For some polynomials, you can factor by grouping pairs of terms that have a common monomial factor. The pattern for factoring by grouping is shown below. ra 1 rb 1 sa 1 sb 5 r(a 1 b) 1 s(a 1 b) 5 (r 1 s)(a 1 b) E XAMPLE 3 Factor by grouping AVOID ERRORS An epression is not factored completely until all factors, such as , cannot be factored further. Factor the polynomial completely ( 2 3) 2 16( 2 3) Factor by grouping. 5 ( )( 2 3) Distributive property 5 ( 1 4)( 2 4)( 2 3) Difference of two squares 354 Chapter 5 Polynomials and Polynomial Functions

19 QUADRATIC FORM An epression of the form au 2 1 bu 1 c, where u is any epression in, is said to be in quadratic form. The factoring techniques you studied in Chapter 4 can sometimes be used to factor such epressions. E XAMPLE 4 Factor polynomials in quadratic form IDENTIFY QUADRATIC FORM The epression is in quadratic form because it can be written as u where u Factor completely: (a) and (b) 2p p p 2. a (4 2 ) Write as difference of two squares. 5 ( )( ) Difference of two squares 5 ( )(2 1 3)(2 2 3) Difference of two squares b. 2p p p 2 5 2p 2 (p 6 1 5p 3 1 6) Factor common monomial. 5 2p 2 (p 3 1 3)(p 3 1 2) Factor trinomial in quadratic form. GUIDED PRACTICE for Eamples 3 and 4 Factor the polynomial completely g t t t 2 SOLVING POLYNOMIAL EQUATIONS In Chapter 4, you learned how to use the zero product property to solve factorable quadratic equations. You can etend this technique to solve some higher-degree polynomial equations. E XAMPLE 5 Standardized Test Practice What are the real-number solutions of the equation ? A 0, 1, 3, 5 B 21, 0, 1 C 0, 1, Ï } 5 D 2 Ï } 5, 21, 0, 1, Ï } 5 Solution Write original equation. AVOID ERRORS Do not divide each side of an equation by a variable or a variable epression, such as 3. Doing so will result in the loss of solutions Write in standard form. 3( ) 5 0 Factor common monomial. 3( 2 2 1)( 2 2 5) 5 0 Factor trinomial. 3 ( 1 1)( 2 1)( 2 2 5) 5 0 Difference of two squares 5 0, 5 21, 5 1, 5 Ï } 5, or 5 2 Ï } 5 Zero product property c The correct answer is D. A B C D GUIDED PRACTICE for Eample 5 Find the real-number solutions of the equation Factor and Solve Polynomial Equations 355

20 E XAMPLE 6 Solve a polynomial equation CITY PARK You are designing a marble basin that will hold a fountain for a city park. The basin s sides and bottom 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 36 cubic feet of water? ANOTHER WAY For alternative methods to solving the problem in Eample 6, turn to page 360 for the Problem Solving Workshop. Solution Volume (cubic feet) 5 Interior length (feet) p Interior width (feet) p Interior height (feet) 36 5 (2 2 2) p ( 2 2) p ( 2 1) 36 5 (2 2 2)( 2 2)( 2 1) Write equation Write in standard form ( 2 4) 1 10( 2 4) Factor by grouping. 0 5 ( )( 2 4) Distributive property c The only real solution is 5 4. The basin is 8 ft long, 4 ft wide, and 4 ft high. GUIDED PRACTICE for Eample WHAT IF? In Eample 6, what should the basin s dimensions be if it is to hold 128 cubic feet of water and have outer length 6, width 3, and height? 5.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 7, 23, and 61 5 STANDARDIZED TEST PRACTICE Es. 2, 9, 41, 63, and VOCABULARY The epression is in? form because it can be written as 2u 2 1 5u 2 3 where u WRITING What condition must the factorization of a polynomial satisfy in order for the polynomial to be factored completely? EXAMPLE 1 on p. 353 for Es. 3 9 MONOMIAL FACTORS Factor the polynomial completely b b 2 5. c 3 1 9c c 6. z 3 2 6z z 7. 3y y m m 4 1 9m 3 9. MULTIPLE CHOICE What is the complete factorization of ? A 2 3 ( 1 2)( 2 2)( 2 1 4) B 2 3 ( 2 1 2)( 2 2 2) C 2 3 ( 2 1 4) 2 D 2 3 ( 1 2) 2 ( 2 2) Chapter 5 Polynomials and Polynomial Functions

21 EXAMPLE 2 on p. 354 for Es EXAMPLE 3 on p. 354 for Es EXAMPLE 4 on p. 355 for Es EXAMPLE 5 on p. 355 for Es SUM OR DIFFERENCE OF CUBES Factor the polynomial completely y m n a c w z FACTORING BY GROUPING Factor the polynomial completely y 3 2 7y 2 1 4y n 3 1 5n 2 2 9n m 3 2 m 2 1 9m s s 2 2 s c 3 1 8c 2 2 9c 2 18 QUADRATIC FORM Factor the polynomial completely a 4 1 7a s 4 2 s z 5 2 2z m m 4 1 m ERROR ANALYSIS Describe and correct the error in finding all real-number solutions (2 1 3)( ) } ( ) or 5 4 SOLVING EQUATIONS Find the real-number solutions of the equation. 32. y 3 2 5y s s 34. g 3 1 3g 2 2 g m 3 1 6m 2 2 4m w w z z b b b p p MULTIPLE CHOICE What are the real-number solutions of the equation ? A 21, 0, 3 B 23, 0, 3 C 23, 0, 1 } 3, 3 D 23, 2 1 } 3, 0, 3 CHOOSING A METHOD Factor the polynomial completely using any method n 4 2 4n b b a a a c c c d d y y y z 5 2 3z z 1 48 GEOMETRY Find the possible value(s) of. 51. Area Volume Volume 5 125π CHOOSING A METHOD Factor the polynomial completely using any method y ac 2 1 bc 2 2 7ad 2 2 bd n 2 2 n CHALLENGE Factor a 5 b 2 2 a 2 b 4 1 2a 4 b 2 2ab 3 1 a 3 2 b 2 completely. 5.4 Factor and Solve Polynomial Equations 357

22 PROBLEM SOLVING EXAMPLE 6 on p. 356 for Es ARCHAEOLOGY At the ruins of Caesarea, archaeologists discovered a huge hydraulic concrete block with a volume of 945 cubic meters. The block s dimensions are meters high by meters long by meters wide. What is the height of the block? LEBANON Caesarea SYRIA EGYPT ISRAEL JORDAN 59. CHOCOLATE MOLD You are designing a chocolate mold shaped like a hollow rectangular prism for a candy manufacturer. The mold must have a thickness of 1 centimeter in all dimensions. The mold s outer dimensions should also be in the ratio 1: 3: 6. What should the outer dimensions of the mold be if it is to hold 112 cubic centimeters of chocolate? 60. MULTI-STEP PROBLEM A production crew is assembling a three-level platform inside a stadium for a performance. The platform has the dimensions shown in the diagrams, and has a total volume of 1250 cubic feet a. Write Epressions What is the volume, in terms of, of each of the three levels of the platform? b. Write an Equation Use what you know about the total volume to write an equation involving. c. Solve Solve the equation from part (b). Use your solution to calculate the dimensions of each of the three levels of the platform. 61. SCULPTURE Suppose you have 250 cubic inches of clay with which to make a sculpture shaped as a rectangular prism. You want the height and width each to be 5 inches less than the length. What should the dimensions of the prism be? 62. MANUFACTURING A manufacturer wants to build a rectangular stainless steel tank with a holding capacity of 670 gallons, or about cubic feet. The tank s walls will be one half inch thick, and about 6.42 cubic feet of steel will be used for the tank. The manufacturer wants the outer dimensions of the tank to be related as follows: The width should be 2 feet less than the length. The height should be 8 feet more than the length. What should the outer dimensions of the tank be? WORKED-OUT SOLUTIONS 358 Chapter 5 Polynomials on p. WS1 and Polynomial Functions 5 STANDARDIZED TEST PRACTICE

23 63. SHORT RESPONSE A platform shaped like a rectangular prism has dimensions 2 2 feet by feet by feet. Eplain why the volume of the platform cannot be } 7 cubic feet EXTENDED RESPONSE In 2000 B.C., the Babylonians solved polynomial equations using tables of values. One such table gave values of y 3 1 y 2. To be able to use this table, the Babylonians sometimes had to manipulate the equation, as shown below. a 3 1 b 2 5 c a 3 3 } 1 a2 2 b 3 } b 2 1 a } b } a 5 a2 c } b 3 b a2 c } b 3 Original equation Multiply each side by a2 } b 3. Rewrite cubes and squares. They then found a2 c } b in the 3 y3 1 y 2 column of the table. Because the corresponding y-value was y 5 } a by, they could conclude that 5 } b a. a. Calculate y 3 1 y 2 for y 5 1, 2, 3,..., 10. Record the values in a table. b. Use your table and the method described above to solve c. Use your table and the method described above to solve d. How can you modify the method described above for equations of the form a 4 1 b 3 5 c? 65. CHALLENGE Use the diagram to complete parts (a) (c). a. Eplain why a 3 2 b 3 is equal to the sum of the volumes of solid I, solid II, and solid III. b. Write an algebraic epression for the volume of each of the three solids. Leave your epressions in factored form. c. Use the results from parts (a) and (b) to derive the factoring pattern for a 3 2 b 3 given on page 354. II b I a b III b a a MIXED REVIEW Graph the function. 66. f() (p. 123) 67. y 5 1 } (p. 236) 68. y 5 3( 1 4) (p. 245) 69. f() (p. 337) Graph the inequality in a coordinate plane. (p. 132) 70. y y > y < y y } } 2 y > 5 PREVIEW Prepare for Lesson 5.5 in Es Use synthetic substitution to evaluate the polynomial function for the given value of. (p. 337) 76. f() ; f() ; f() ; f() ; 5 4 EXTRA PRACTICE for Lesson 5.4, p ONLINE Factor and QUIZ Solve Polynomial at classzone.com Equations 359

24 LESSON 5.4 Using ALTERNATIVE METHODS Another Way to Solve Eample 6, page 356 MULTIPLE REPRESENTATIONS In Eample 6 on page 356, you solved a polynomial equation by factoring. You can also solve a polynomial equation using a table or a graph. P ROBLEM CITY PARK You are designing a marble basin that will hold a fountain for a city park. The basin s sides and bottom 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 36 cubic feet of water? M ETHOD 1 Using a Table One alternative approach is to write a function for the volume of the basin and make a table of values for the function. Using the table, you can find the value of that makes the volume of the basin 36 cubic feet. STEP 1 Write the function. From the diagram, you can see that the volume y of water the basin can hold is given by this function: y 5 (2 2 2)( 2 2)( 2 1) STEP 2 Make a table of values for the function. Use only positive values of because the basin s dimensions must be positive. STEP 3 Identify the value of for which y The table shows that y 5 36 when 5 4. X Y1= Y1 X Y1= Y1 c The volume of the basin is 36 cubic feet when is 4 feet. So, the outer dimensions of the basin should be as follows: Length feet Width feet Height feet 360 Chapter 5 Polynomials and Polynomial Functions

25 M ETHOD 2 Using a Graph Another approach is to make a graph. You can use the graph to find the value of that makes the volume of the basin 36 cubic feet. STEP 1 Write the function. From the diagram, you can see that the volume y of water the basin can hold is given by this function: y 5 (2 2 2)( 2 2)( 2 1) STEP 2 Graph the equations y 5 36 and y 5 ( 2 1)(2 2 2)( 2 2). Choose a viewing window that shows the intersection of the graphs. STEP 3 Identify the coordinates of the intersection point. On a graphing calculator, you can use the intersect feature. The intersection point is (4, 36). Intersection X=4 Y=36 c The volume of the basin is 36 cubic feet when is 4 feet. So, the outer dimensions of the basin should be as follows: Length feet Width feet Height feet P RACTICE SOLVING EQUATIONS Solve the polynomial equation using a table or using a graph WHAT IF? In the problem on page 360, suppose the basin is to hold 200 cubic feet of water. Find the outer dimensions of the basin using a table and using a graph. 9. PACKAGING A factory needs a bo that has a volume of 1728 cubic inches. The width should be 4 inches less than the height, and the length should be 6 inches greater than the height. Find the dimensions of the bo using a table and using a graph. 10. AGRICULTURE From 1970 to 2002, the average yearly pineapple consumption P (in pounds) per person in the United States can be modeled by the function P() where is the number of years since In what year was the pineapple consumption about 9.97 pounds per person? Solve the problem using a table and a graph. Using Alternative Methods 361