5.3 Polynomials and Polynomial Functions

Transcription

1 70 CHAPTER 5 Eponents, Polnomials, and Polnomial Functions 5. Polnomials and Polnomial Functions S Identif Term, Constant, Polnomial, Monomial, Binomial, Trinomial, and the Degree of a Term and of a Polnomial. Define Polnomial Functions. Review Combining Like Terms. 4 Add Polnomials. 5 Subtract Polnomials. 6 Recognize the Graph of a Polnomial Function from the Degree of the Polnomial. Identifing Polnomial Terms and Degrees of Terms and Polnomials A term is a number or the product of a number and one or more variables raised to powers. The numerical coefficient, or simpl the coefficient, is the numerical factor of a term. Term Numerical Coefficient of Term z aor 7 9 b 7 If a term contains onl a number, it is called a constant term, or simpl a constant. A polnomial is a finite sum of terms in which all variables are raised to nonnegative integer powers and no variables appear in an denominator. Polnomials Not Polnomials z Negative integer eponent Variable in denominator Helpful Hint We usuall write answers that are polnomials in one variable in descending order. A polnomial that contains onl one variable is called a polnomial in one variable. For eample, is a polnomial in. This polnomial in is written in descending order since the terms are listed in descending order of the variable s eponents. (The term 7 can be thought of as 7 0.) The following eamples are polnomials in one variable written in descending order a a + 4a A monomial is a polnomial consisting of one term. A binomial is a polnomial consisting of two terms. A trinomial is a polnomial consisting of three terms. Monomials Binomials Trinomials a z - z B definition, all monomials, binomials, and trinomials are also polnomials. Each term of a polnomial has a degree. Degree of a Term The degree of a term is the sum of the eponents on the variables contained in the term.

2 Section 5. Polnomials and Polnomial Functions 7 EXAMPLE Find the degree of each term. a. b. - 5 c. d. z e. 5.7 Solution a. The eponent on is, so the degree of the term is. b. The eponent on is 5, so the degree of the term is 5. (Recall that the degree is the sum of the eponents on onl the variables.) c. The degree of, or, is. d. The degree is the sum of the eponents on the variables, or + + = 6. e. The degree of 5.7, which can be written as 5.7 0, is 0. Find the degree of each term. a. 4 5 b. -4 c. z d. 65a b 7 c e. 6 From the preceding eample, we can sa that the degree of a constant is 0. Also, the term 0 has no degree. Each polnomial also has a degree. Degree of a Polnomial The degree of a polnomial is the largest degree of all its terms. EXAMPLE Find the degree of each polnomial and indicate whether the polnomial is also a monomial, binomial, or trinomial. Polnomial Degree Classification a Trinomial b. -z + + = Monomial c Binomial Find the degree of each polnomial and indicate whether the polnomial is also a monomial, binomial, or trinomial. Polnomial Degree Classification a b. 9abc c EXAMPLE Solution The degree of each term is Find the degree of the polnomial T T T T Degree: 4 0

3 7 CHAPTER 5 Eponents, Polnomials, and Polnomial Functions The largest degree of an term is 4, so the degree of this polnomial is 4. Find the degree of the polnomial Defining Polnomial Functions At times, it is convenient to use function notation to represent polnomials. For eample, we ma write P to represent the polnomial In smbols, this is P = This function is called a polnomial function because the epression is a polnomial. Helpful Hint Recall that the smbol P does not mean P times. It is a special smbol used to denote a function. EXAMPLE 4 If P = - - 5, find the following. a. P b. P - Solution a. Substitute for in P = and simplif. P = P = = -4 b. Substitute - for in P = and simplif. P = P - = = 4 If P = , find the following. a. P - b. P Man real-world phenomena are modeled b polnomial functions. If the polnomial function model is given, we can often find the solution of a problem b evaluating the function at a certain value. EXAMPLE 5 Finding the Height of an Object The world s highest bridge, the Millau Viaduct in France, is 5 feet above the River Tarn. An object is dropped from the top of this bridge. Neglecting air resistance, the height of the object at time t seconds is given b the polnomial function Pt = -6t + 5. Find the height of the object when t = second and when t = 8 seconds.

4 Section 5. Polnomials and Polnomial Functions 7 Solution To find the height of the object at second, we find P. Pt = -6t + 5 P = P = 09 When t = second, the height of the object is 09 feet. To find the height of the object at 8 seconds, we find P8. Pt = -6t + 5 P8 = P8 = 0 When t = 8 seconds, the height of the object is 0 feet. Notice that as time t increases, the height of the object decreases. 5 The largest natural bridge is in the canons at the base of Navajo Mountain, Utah. From the base to the top of the arch, it measures 90 feet. Neglecting air resistance, the height of an object dropped off the bridge is given b the polnomial function Pt = -6t + 90 at time t seconds. Find the height of the object at time t = 0 second and t = seconds. Combining Like Terms Review Before we add polnomials, recall that terms are considered to be like terms if the contain eactl the same variables raised to eactl the same powers. Like Terms Unlike Terms -5, - 4, 7 z, -z, - To simplif a polnomial, we combine like terms b using the distributive propert. For eample, b the distributive propert, = = Helpful Hint These two terms are unlike terms. The cannot be combined. EXAMPLE 6 Simplif b combining like terms. a b Solution B the distributive propert, a = = -5-6 b. Use the associative and commutative properties to group together like terms; then combine = = = 8 - ()* " 6 Simplif b combining like terms. a b. 4ab - 5b + ab + b 4 Adding Polnomials Now we have reviewed the necessar skills to add polnomials.

5 74 CHAPTER 5 Eponents, Polnomials, and Polnomial Functions Adding Polnomials To add polnomials, combine all like terms. EXAMPLE 7 Add and Solution = Group like terms. = Combine like terms. 7 Add and EXAMPLE 8 Add. a b. a - b + a a + b + 5 Solution a. To add, remove the parentheses and group like terms = = Group like terms. = Combine like terms. b. a - b + a a + b + 5 = a - b + a a + b + 5 = a - b + b + a + a Group like terms. = a + a Combine like terms. 8 Add. a. a 4 b - 5ab ab - b Sometimes it is more convenient to add polnomials verticall. To do this, line up like terms beneath one another and add like terms. An eample is shown later in this section. 5 Subtracting Polnomials The definition of subtraction of real numbers can be etended to appl to polnomials. To subtract a number, we add its opposite. a - b = a + -b Likewise, to subtract a polnomial, we add its opposite. In other words, if P and Q are polnomials, then P - Q = P + -Q The polnomial -Q is the opposite, or additive inverse, of the polnomial Q. We can find -Q b writing the opposite of each term of Q.

6 Section 5. Polnomials and Polnomial Functions 75 CONCEPT CHECK Which polnomial is the opposite of ? a b c d Subtracting Polnomials To subtract a polnomial, add its opposite. For eample, To subtract, change the signs; then add. ( +4-7)-( --5)=( +4-7)+( ++5) = =6- Combine like terms. EXAMPLE 9 Subtract: z 5 - z + z - -z 4 + z + z Solution To subtract, add the opposite of the second polnomial to the first polnomial. z 5 - z + z - -z 4 + z + z = z 5 - z + z + z 4 - z - z = z 5 + z 4 - z - z + z - z = z 5 + z 4 - z - z Add the opposite of the polnomial being subtracted. Group like terms. Combine like terms. 9 Subtract: a 4-7a a 4 + 8a -. CONCEPT CHECK Wh is the following subtraction incorrect? 7z z - 4 = 7z z - 4 = 4z - 9 EXAMPLE 0 Subtract from 0-7. Solution If we subtract from 8, the difference is 8 - = 6. Notice the order of the numbers and then write Subtract from 0-7 as a mathematical epression. Answers to Concept Checks: b; With parentheses removed, the epression should be 7z z + 4 = 4z = Remove parentheses. = Combine like terms. 0 Subtract from - 7.

7 76 CHAPTER 5 Eponents, Polnomials, and Polnomial Functions To add or subtract polnomials verticall, just remember to line up like terms. For eample, perform the subtraction, from Eample 0, verticall. Add the opposite of the second polnomial is equivalent to Polnomial functions, like polnomials, can be added, subtracted, multiplied, and divided. For eample, if then P = + + P = + + = + + Use the distributive propert. Also, if Q = 5 -, then P + Q = = 6 +. A useful business and economics application of subtracting polnomial functions is finding the profit function P when given a revenue function R and a cost function C. In business, it is true that profit = revenue - cost, or P = R - C For eample, if the revenue function is R = 7 and the cost function is C = , then the profit function is P = R - C or P = Substitute R = 7 P = and C = Problem-solving eercises involving profit are in the eercise set Recognizing Graphs of Polnomial Functions from Their Degree In this section, we reviewed how to find the degree of a polnomial. Knowing the degree of a polnomial can help us recognize the graph of the related polnomial function. For eample, we know from Section. that the graph of the polnomial function f = is a parabola as shown to the left. The polnomial has degree. The graphs of all polnomial functions of degree will have this same general shape opening upward, as shown, or downward. Graphs of polnomial functions of degree or will, in general, resemble one of the graphs shown net. General Shapes of Graphs of Polnomial Functions Degree Coefficient of is a positive number. Coefficient of is a negative number.

8 Section 5. Polnomials and Polnomial Functions 77 Degree or or Coefficient of is a positive number. Coefficient of is a negative number. EXAMPLE Determine which of the following graphs most closel resembles the graph of f = A B C D Solution The degree of f is, which means that its graph has the shape of B or D. The coefficient of is 5, a positive number, so the graph has the shape of B. Determine which of the following graphs most closel resembles the graph of f = -. A B C D Graphing Calculator Eplorations A graphing calculator ma be used to visualize addition and subtraction of polnomials in one variable. For eample, to visualize the following polnomial subtraction statement = - + graph both Y = Left side of equation and Y = - + Right side of equation on the same screen and see that their graphs coincide. (Note: If the graphs do not coincide, we can be sure that a mistake has been made in combining polnomials or in calculator kestrokes. If the graphs appear to coincide, we cannot be sure that our work is correct. This is because it is possible for the graphs to differ so slightl that we do not notice it.)

9 78 CHAPTER 5 Eponents, Polnomials, and Polnomial Functions The graphs of Y and Y are shown. The graphs appear to coincide, so the subtraction statement = - + appears to be correct. Perform the indicated operations. Then visualize b using the procedure described above Vocabular, Readiness & Video Check Use the choices below to fill in each blank. Not all choices will be used. monomial trinomial like degree coefficient binomial polnomial unlike variables term. The numerical factor of a term is the.. A(n) is a finite sum of terms in which all variables are raised to nonnegative integer powers and no variables appear in an denominator.. A(n) is a polnomial with terms. 4. A(n) is a polnomial with term. 5. A(n) is a polnomial with terms. 6. The degree of a term is the sum of the eponents on the in the term. 7. The of a polnomial is the largest degree of all its terms. 8. terms contain the same variables raised to the same powers. Martin-Ga Interactive Videos Watch the section lecture video and answer the following questions. 9. In Eample, wh is the degree of each term found when this is not asked for? 0. From Eample 4, finding the value of a polnomial function at a given replacement value is similar to what? See Video How is the polnomial in Eample 5 simplified?. From Eample 6, how do ou add polnomials?. From Eamples 7 and 8, how do ou change a polnomial subtraction problem into an equivalent addition problem? 6 4. Which of the basic graph shapes, A, B, C, or D, in Eamples 9 and 0 most closel resembles the graph of f = ? Eplain.

10 Section 5. Polnomials and Polnomial Functions Eercise Set Find the degree of each term. See Eample z z ab c r st Find the degree of each polnomial and indicate whether the polnomial is a monomial, binomial, trinomial, or none of these. See Eamples and abc If P = + + and Q = 5 -, find the following. See Eample P7 0. Q4. Q -0. P -4. Qa 4 b 4. Pa b Refer to Eample 5 for Eercises 5 through Find the height of the object at t = seconds. 6. Find the height of the object at t = 4 seconds. 7. Find the height of the object at t = 6 seconds. 8. Approimate (to the nearest second) how long it takes before the object hits the ground. (Hint: The object hits the ground when P = 0.) Simplif each polnomial b combining like terms. See Eample a + 8ab - b + 4a - ab - b Add. See Eamples 7 and and and z z Subtract. See Eamples 9 and Subtract 6 - from Subtract from Subtract a b from a b. 54. Subtract a b from a b. MIXED Perform indicated operations and simplif. See Eamples 6 through b + a b + 6a a + 7b a + 6b ab - 0a b + 6b - 8a - 0a b - 6b

11 80 CHAPTER 5 Eponents, Polnomials, and Polnomial Functions Subtract + 7 from the sum of and Subtract from the sum of and a b - a b 80. a b - a b Use the information below to solve Eercises 8 and 8. The surface area of a rectangular bo is given b the polnomial function SA = HL + LW + HW and is measured in square units. In business, surface area is often calculated to help determine cost of materials. H W 8. A rectangular bo is to be constructed to hold a new camcorder. The bo is to have dimensions 5 inches b 4 inches b 9 inches. Find the surface area of the bo. 8. Suppose it has been determined that a bo of dimensions 4 inches b 4 inches b 8.5 inches can be used to contain the camcorder in Eercise 8. Find the surface area of this bo and calculate the square inches of material saved b using this bo instead of the bo in Eercise A projectile is fired upward from the ground with an initial velocit of 00 feet per second. Neglecting air resistance, the height of the projectile at an time t can be described b the polnomial function Pt = -6t + 00t. Find the height of the projectile at each given time. a. t = second b. t = seconds c. t = 0 seconds d. t = 4 seconds L e. Eplain wh the height increases and then decreases as time passes. f. Approimate (to the nearest second) how long before the object hits the ground. 84. A worker at the Hoover Dam bpass bridge was spending his lunch break tossing a football. He threw the ball upward with an initial velocit of 0 feet per second but missed the ball, and it went over his head and down toward the river. The height of the football above the Colorado River at an time t can be described b the polnomial function Pt = -6t + 0t Find the height of the football at each given time. a. t = 0 seconds b. t = second c. t = seconds d. t = 5 seconds e. Eplain wh the height increases and then decreases as time passes. f. Approimate (to the nearest second) how long before the football lands in the river. 85. The polnomial function P = 45-00,000 models the relationship between the number of computer briefcases that a compan sells and the profit the compan makes, P. Find P4000, the profit from selling 4000 computer briefcases. 86. The total cost (in dollars) for MCD, Inc., Manufacturing Compan to produce blank CDs per week is given b the polnomial function C = Find the total cost of producing 0,000 CDs per week. 87. The total revenues (in dollars) for an art suppl compan to sell boes of colored pencils per week over the Internet is given b the polnomial function R =. Find the total revenue from selling 500 boes of colored pencils. 88. The total revenues (in dollars) for MCD, Inc., Manufacturing Compan to sell blank CDs per week is given b the polnomial function R = 0.9. Find the total revenue from selling 0,000 CDs per week. Match each equation with its graph. See Eample. 89. f = h = g = F = A B

12 Section 5. Polnomials and Polnomial Functions 8 C REVIEW AND PREVIEW D Multipl. See Section z CONCEPT EXTENSIONS Solve. See the Concept Checks in this section. 97. Which polnomial(s) is the opposite of 8-6? a b c d Which polnomial(s) is the opposite of ? a b c d Correct the subtraction = Correct the addition z 5 + z + z - z 5-0z + z Find each perimeter. 09. ( 5) units ( ) units = = = Write a function, P, so that P0 = Write a function, R, so that R =. 0. In our own words, describe how to find the degree of a term. 04. In our own words, describe how to find the degree of a polnomial. Perform the indicated operations a - a a - 5 a a - 4 a a - a (z ) units (z 4z ) units (z z) units If P = +, Q = 4-6 +, and R = 5-7, find the following.. P + Q. R + P. Q - R 4. P - Q 5. [Q] - R 6. -5[P] - Q 7. [R] + 4[P] 8. [Q] + 7[R] If P is the polnomial given, find a. Pa, b. P -, and c. P + h. 9. P = - 0. P = 8 +. P = 4. P = -4. P = 4-4. P = - 5. The function f = can be used to approimate the amount of restaurant food-and-drink sales, where is the number of ears since 970 and f() or is the sales (in billions of dollars.) a. Approimate the restaurant food-and-drink sales in 005. b. Approimate the restaurant food-and-drink sales in 00. c. Use this function to estimate the restaurant food-anddrink sales in 05. d. From parts (a), (b), and (c), determine whether the restaurant food-and-drink sales is increasing at a stead rate. Eplain wh or wh not. Sales (billions of dollars) Restaurant Food-and-Drink Sales Year Source: National Restaurant Association

13 8 CHAPTER 5 Eponents, Polnomials, and Polnomial Functions 6. The function f () = can be used to approimate the total cheese production in the United States from 000 to 009, where is the number of ears after 000 and is pounds of cheese (in billions). Round answers to the nearest hundredth of a billion. (Source: National Agricultural Statistics Service, USDA) a. Approimate the number of pounds of cheese produced in the United States in 000. b. Approimate the number of pounds of cheese produced in the United States in 005. c. Use this function to estimate the pounds of cheese produced in the United States in 05. d. From parts (a), (b), and (c), determine whether the number of pounds of cheese produced in the United States is increasing at a stead rate. Eplain wh or wh not. Pounds of Cheese (in billions) Total U.S. Cheese Production Years since Multipling Polnomials S Multipl Two Polnomials. Multipl Binomials. Square Binomials. 4 Multipl the Sum and Difference of Two Terms. 5 Multipl Three or More Polnomials. 6 Evaluate Polnomial Functions. Multipling Two Polnomials Properties of real numbers and eponents are used continuall in the process of multipling polnomials. To multipl monomials, for eample, we appl the commutative and associative properties of real numbers and the product rule for eponents. EXAMPLE Multipl. a. 5 6 b. 7 4 z 4 - z 5 Solution Group like bases and appl the product rule for eponents. a. 5 6 = 5 6 = 0 9 b. 7 4 z 4 - z 5 = 7-4 z 4 z 5 = -7 5 z 9 Multipl. a. 4 b. -5m 4 np -8mnp 5 Helpful Hint See Sections 5. and 5. to review eponential epressions further. To multipl a monomial b a polnomial other than a monomial, we use an epanded form of the distributive propert. ab + c + d + g + z = ab + ac + ad + g + az Notice that the monomial a is multiplied b each term of the polnomial. EXAMPLE Multipl. a. 5-4 b c Solution Appl the distributive propert. a. (5-4)=(5)+( 4) = 0-8 Use the distributive propert. Multipl.