Chapter 1 POLYNOMIAL FUNCTIONS

Transcription

1 Chapter POLYNOMIAL FUNCTIONS Have you ever wondered how computer graphics software is able to so quickly draw the smooth, life-like faces that we see in video games and animated movies? Or how in architectural projects builders compensate for the fact that a horizontal beam, fied in position at both ends, will bend under its own weight? Can you imagine how computers mould automotive body panels? Believe it or not, all three tasks are possible thanks to polynomials! Polynomials are composed by applying addition, subtraction, and multiplication to numbers and variables. The information needed to perform certain tasks like the ones listed above is reduced to the polynomial segments between key points. Much like words in language, polynomials are the vocabulary of algebra, and, as such, they are used in a wide variety of applications by designers, engineers, and others. Calculus, the study of motion and rates of change, requires a clear understanding of polynomials, so we ll begin our study there. CHAPTER EXPECTATIONS In this chapter, you will determine properties of the graphs of polynomial functions, Section. sketch the graph of a polynomial function, Section. describe the nature of change in polynomial functions, Section. determine an equation to represent a given graph of a polynomial function, Career Link understand the Remainder and Factor Theorems, Section.3,.4

2 Review of Prerequisite Skills Before beginning your study of Polynomial Functions, you may wish to review the following factoring methods that you learned in previous courses. Common Factor 4 8 4( ) Grouping By grouping terms together it is often possible to factor the grouped terms. Factor fully a c ay cy (a c) (ay cy) (a c) y(a c) (a c)( y) Trinomial Factoring Factor fully Solution (by decomposition) Solution (by inspection) ( )(3 4) 3( ) 4( ) ( )(3 4) Factor 0. Solution Create a chart using factors of and Notice that what looks like a lot of work can be greatly simplified when numbers in the upper right that have common factors with, 6, and 4 are crossed out. The reduced chart is CHAPTER

3 From the numbers that remain, we see that 4 (4) 6, and gives 6 5. Therefore, 0 (4 5)(3 4). Difference of Squares Because (a b)(a b) a b, it is always possible to factor the difference between two perfect squares. 6 8 (4 9)(4 9) Special Cases Sometimes by grouping terms, the difference between squares can be created. a p a (a a ) p (a ) p [(a ) p][(a ) p] (a p)(a p) Eercise. Factor fully. a. p pr r b. 6n 8n c. 9u 30u 5 d. v 4v 3 e. w 3w f. 3k 7k g. 7y 5y h i. 3v v 0. Factor fully. a. 5 y b. m p c. 6r d. 49m 64 e. p r 00 f. 3 48y g. ( n) 9 h. 49u ( y) i Factor fully. a. k p ky py b. f gy g fy c. h 3 h h d. d ( d) e. 4y 4yz z f. y z z REVIEW OF PREREQUISITE SKILLS 3

4 4. Factor fully. a. 4 6 b. 8s 8st 0t c. y (r n) d. 8 4m 80m e f. y 3 y 5y 5 g. 60y 0y 0 h i Factor fully. a. 36( y) 5(u y) b. g( ) g g c. y 5 y 4 y 3 y y d. n 4 n w w 4 e. 9( y z) 6( y z) f. 8u (u ) u(u ) 3(u ) g. p p y yz z h. 9y 4 y 4 i. ab (an bm) mn j. 4 CHAPTER

5 CHAPTER : MODELLING WATER DEMAND Imagine if you woke up one morning looking forward to a shower only to have your mom tell you the local water utility ran out of water because they made a mistake in predicting demand. That does not happen, in part, because water utilities develop reliable mathematical models that accurately predict water demand. Of particular use in mathematical modelling are the polynomial functions that you will investigate in this chapter. You are already familiar with two classes of polynomials: the linear (y m b) and the quadratic (y a b c). You can find polynomial mathematical models in a multitude of places, from computers (e.g., Internet encryption), to business (e.g., the mathematics of investment), to science (e.g., population dynamics of wildlife). Case Study Municipal Engineer/Technologist Civil Engineers and Technologists frequently model the relationship between municipal water demand and time of day to ensure that water supply meets demand plus a factor of safety for fire flows. Water demand data for a city with a population of is presented in the table below. DISCUSSION QUESTIONS investigate Water Demand for Blueborough, Ontario Time of Day t Water Demand (in hours) (in cubic metres per hour) 3: : : : : : : Plot a rough sketch of the data in the table above. What kind of relationship, if any, does the data show? Remember that you have been investigating linear, quadratic, rational, and periodic functions. Does the hour-to-hour trend in the data make sense? Eplain.. Sketch the water demand over a 4-h period for your community. Use an average daily demand of 600 L per capita and a peak hourly flow of about.5 times the average hourly flow. Eplain the peaks and valleys. 3. Find out how much water costs in your community and estimate the cost per hour of operating your community s water distribution system at the peak flow rate determined in Question. At the end of this chapter you will develop and utilize a mathematical model for the data presented in this case study. CAREER LINK 5

6 Section. Graphs of Polynomial Functions The graph of a linear function of the form f () a b has either one -intercept or no -intercepts. Function Graph Number of -intercepts f () y f() f () y f() No -intercepts By graphing a quadratic function of the form f () a b c, a 0, we can determine the number of -intercepts. Each -intercept indicates a real root of the corresponding quadratic equation. Function Graph Number of -intercepts f () 7 0 y f() chart continued 6 CHAPTER

7 f () 6 9 y When a curve touches the -ais, there are two equal roots for the corresponding f() f () f() y There are no real roots. INVESTIGATION : CUBIC FUNCTIONS t chnology e APPENDIX P. 47. Use a graphing calculator or a computer to graph each of the following cubic functions. Sketch each of the graphs in your notebook so that you can make observations about the shapes of the graphs and list the number of -intercepts. a. y 3 b. y 3 c. y 3 d. y 3 3 e. y f. y 3 3 g. y h. y i. y ( )( )(3 ). From your observations, list the possible numbers of real roots for a cubic equation. 3. a. Eplain how you would graph the cubic function y ( )( 3)( 4) without using a graphing calculator. b. Draw a sketch of the function in part a.. GRAPHS OF POLYNOMIAL FUNCTIONS 7

8 4. Sketch two possible general shapes for the graph of a cubic function that has a coefficient of 3 that is positive. 5. For the functions in Question, change the coefficient of 3 from positive to negative and redraw the graphs. For eample, y 3 changes to y 3. What observation do you make for the general shape of the graph of a cubic function that has a coefficient of 3 that is negative? INVESTIGATION : QUARTIC FUNCTIONS t chnology e. Use a graphing calculator or a computer to graph each of the following quartic functions. Sketch each of the graphs in your notebook so that you can make observations about the shapes of the graphs and list the number of -intercepts. a. y 4 b. y 4 4 c. y d. y e. y f. y ( )( )( 3)( 3). From your observations, list the possible numbers of real roots for a quartic equation. 3. a. Eplain how you would graph the quartic function y ( 3)( )( )( 4) without using a graphing calculator. b. Draw a sketch of the function in part a. 4. Sketch two possible general shapes for the graph of a quartic function that has a coefficient of 4 that is positive. 5. For the functions in Question, change the coefficient of 4 from positive to negative and redraw the graphs. For eample, y changes to y What observation do you make for the general shape of the graph of a quartic function that has a coefficient of 4 that is negative? INVESTIGATION 3 t chnology e. Use your graphing calculator to graph each of the following: a. y ( 3) b. y ( )( )( ) c. y ( ) ( ) Based on these graphs, draw a sketch of what you think the graph of y ( )( ) looks like. 8 CHAPTER

9 . Use your graphing calculator to graph each of the following: a. y ( ) 3 b. y ( 3) 3 c. y ( ) ( ) 3 Based on these graphs, draw a sketch of what you think the graph of y ( )( ) 3 looks like. Eercise. Part A Knowledge/ Understanding t chnology e. Check your conclusions about the shape of the graphs of functions by using your graphing calculator to draw each of the following: a. y 3 6 b. y c. y 3 6 d. y e. y ( 3)(3 )( )( 3) f. y ( )( 3)(9 4) g. y h. y Part B Application Communication Thinking/Inquiry/ Problem Solving. Draw a rough sketch (without using your graphing calculator) of each of the following: a. y ( )( ) b. y ( )( )( 3) c. y ( )( 3)( )( 4) d. y ( )( ) 3. a. Draw as many different shapes as possible of a cubic function. b. Draw as many different shapes as possible of a quartic function. 4. You have investigated the general shape of the graphs of cubic and quartic functions. Sketch a possible general shape for the graphs of each of the following: a. A fifth-degree function that has a coefficient of 5 that is (i) positive (ii) negative b. A sith-degree function that has a coefficient of 6 that is (i) positive (ii) negative. GRAPHS OF POLYNOMIAL FUNCTIONS 9

10 Section. Polynomial Functions from Data In earlier courses, you used finite differences as a means of identifying polynomial functions. If we have the right data we can obtain a sequence of first differences, second differences, and so on. The purpose of the investigation in this section is to determine the pattern of finite differences for given polynomials. The table below lists finite differences for the linear function f(). f() f() m m m (m ) m m m m m m The set of first differences of a linear function is constant. INVESTIGATION The purpose of this investigation is to determine the pattern of finite differences for quadratic and cubic functions.. For the function f(), copy and complete the table below, calculating first differences, second differences, and so on, to determine whether or not the sequence of entries becomes constant. f() f() f() * 3 f() first difference second difference third difference 3 m m m m m * f () means second difference. 0 CHAPTER

11 . For the function f() 3, copy and complete the table below, calculating first differences, second differences, and so on, to determine whether or not the sequence of entries becomes constant. f() f() f() * 3 f() first difference second difference third difference 3 m m m m m * f () means second difference. If the set {m, m, m, m, m } describes every set of five consecutive values, can you make a general statement about the pattern of successive finite differences for polynomial functions? EXAMPLE t chnology e Given that the points (,), (, 3), (3, 5), (4, 37), (5, 05), and (6, ) lie on the graph of a polynomial function, determine a possible epression for the function having integer coefficients. Solution Input the data in your graphing calculator as follows:. Select the STAT function and press ENTER to select EDIT mode.. In the L column, input,, 3, 4, 5, 6, and for the L column, input, 3, 5, 37, 05,. LIST STAT 3. Move the cursor to the L 3 column. Select nd for the LIST function. Move the cursor to OPS and then select option 7: List(. 4. Enter L in the List (L ) to obtain the first finite differences for L. 5. Move the cursor to the L 4 column. Repeat steps 3 and 4 to obtain the second finite differences for L 3. Note: Enter L 3 in the List (L 3 ). 6. Move the cursor to the L 5 column. Repeat steps 3 and 4 to obtain the third finite differences for L 3. Note: Enter L 4 in the List (L 4 ).. POLYNOMIAL FUNCTIONS FROM DATA

12 If the first finite difference is constant, then f() is a linear function. If the second finite difference is constant, then f() is a quadratic function. The third finite difference in column L 5 is constant. If f() is a polynomial function, then it must be cubic, of the form f() a 3 b c d. Use the CubicReg function to obtain the following result. The CubicReg function is located in the CALC mode on the STAT key. Note that c.4 0 is a very small number, so let c 0 and the required result is f() A second method, using algebra, is as follows. Let the function be f(). Using differences, we obtain the following: f() f() f() 3 f() From the data, 3 f() is constant. If f() is a polynomial, it must be cubic, therefore f() must be of the form f() a 3 b c d. Using the given ordered pairs, we get f() a b c d f() 8a 4b c d 3 f(3) 7a 9b 3c d 5 3 f(4) 64a 6b 4c d 37 4 CHAPTER

13 Solving these equations, we have 7a 3b c a 5b c a 7b c a b a b a a Substituting into 8 4 b b 6 Substituting into c 4 c 0 Substituting into 6 0 d d 5 Therefore, the function is f() Eercise. Part A t chnology e Knowledge/ Understanding In each of the following, you are given a set of points that lie on the graph of a function. Determine, if possible, the equation of the polynomial function using a graphing calculator or the algebraic method.. (, 0), (, ), (3, ), (4, 0), (5, 4), (6, 0). (, ), (, ), (3, 5), (4, 8), (5, ), (6, 4) 3. (, 4), (, 5), (3, 30), (4, 49), (5, 7), (6, 99) 4. (, 9), (, 0), (3, 7), (4, 0), (5, ), (6, 6) 5. (, ), (, 0), (3, 8), (4, 0), (5, 56), (6, 6) 6. (, 34), (, 4), (3, 38), (4, 6), (5, 30), (6, 06) 7. (, 0), (, 0), (3, 0), (4, 6), (5, 54), (6, 0), (7, 0) 8. (, 4), (, 0), (3, 30), (4, 98), (5, 6), (6, 396) 9. (, ), (, 4), (3, 6), (4, 8), (5, 4), (6, 08), (7, 346) 0. (, ), (, ), (3, 4), (4, 8), (5, 6), (6, 3), (7, 64). POLYNOMIAL FUNCTIONS FROM DATA 3

14 Part B Application. The volume, V, of air in the lungs during a 5 s respiratory cycle is given by a cubic function (with time t as the independent variable). a. The following data was recorded: t (in seconds) V (in litres) t chnology e Thinking/Inquiry/ Problem Solving Determine the cubic function that satisfies this data. b. Using your graphing calculator, find the maimum volume of air in the lungs during the cycle, and find when during the cycle this maimum occurs.. a. The population of a town is given by a polynomial function. Let time, t, be the independent variable, t 0 in 98, and use the data below to determine the function. Year Population b. The town seemed destined to become a ghost town until oil was discovered there and the population started to increase. In what year did this happen? c. If the function continues to describe the population correctly, what will the population be in 030? 4 CHAPTER

15 Section.3 Division of Polynomials Division of polynomials can be done using a method similar to that used to divide whole numbers. Since division of polynomials cannot be done on all calculators, let s first review the division process in arithmetic. EXAMPLE Divide 579 by 8. Solution Step : Divide 8 into 57, obtaining 7. Step : Multiply 8 by 7, obtaining 56. Step 3: Subtract 56 from 57, obtaining. Step 4: Bring down the net digit after 57. Step 5: Repeat steps 4 using the new number, 9. Step 6: Stop when the remainder is less than 8. We can state the results in the form of the division statement Division with polynomials follows the same procedure. When you are performing division, you should write both the divisor and dividend in descending powers of the variable. EXAMPLE Divide 7 0 by. Solution Step : Divide first term of the dividend ( 7 0) by the first term of the divisor [i.e., ]. Step : Multiply (( ) ), placing the terms below those in the dividend of the same power. Step 3: Subtract and bring down the net term. Step 4: Repeat steps 3. Step 5: Stop when the degree of the remainder is less than that of the divisor. We can epress the results as 7 0 ( )( 9) 8. Note: This is of the form dividend divisor quotient remainder or f() d()q() r(). EXAMPLE 3 Perform the following divisions and epress the answers in the form f() d()q() r(). a. ( ) ( 3) b. ( 3 4) ( ).3 DIVISION OF POLYNOMIALS 5

16 Solution a Since the remainder, r(), is of a degree less than that of the divisor, the division is complete ( 3)( 3 5) b. Insert 0 in the function so that every term is present Since the remainder is 0, is a factor of 3 4. The other factor is. 3 4 ()( ) EXAMPLE 4 Perform the following division and epress the answer in the form f() d()q() r(). ( ) ( 3 ). Solution Since the remainder, r() 5 8, is of a lower degree than the divisor, 3, the division is complete ( 3 )(3 7 ) (5 8) EXAMPLE 5 Determine the remainder when is divided by: a. 3 b. 3 6 CHAPTER

17 Solution a b The remainders are equal. Is this always true if a function is divided by p t and t by p? Suppose that f() divided by d() p t produces quotient q() and remainder r(). We can write f() (p t)q() r(). Now f() (p t)q() r() t p p q() r() t p [p q()] r(). t From this it is clear that division by p produces a quotient greater by a factor p than that of division by (p t), but the remainders are the same. Eercise.3 Part A. Perform each of the following divisions and epress the result in the form dividend divisor quotient remainder. a. 7 5 b. 4 7 c. 73 d e f Communication. a. In Question a, eplain why 5 is not a factor of 7. b. In Question b, eplain why 7 is a factor of 4. c. In Questions d and f, what other divisor is a factor of the dividend?.3 DIVISION OF POLYNOMIALS 7

18 Communication 3. Eplain the division statement f() d()q() r() in words. Part B Knowledge/ Understanding 4. For f() ( )( 3 ) 5, a. identify the linear divisor d(). b. identify the quotient q(). c. identify the remainder r(). d. determine the dividend f(). 5. When a certain polynomial is divided by 3, its quotient is 5 7 and its remainder is 5. What is the polynomial? Application 6. When a certain polynomial is divided by, its quotient is and its remainder is. What is the polynomial? 7. In each of the following, divide f() by d(), obtaining quotient q() and remainder r. Write your answers in the form f() d()q() r(). a. ( 3 3 ) ( ) b. ( ) ( ) c. ( ) ( 3) d. (3 3 6) ( ) e. (3 4) ( 4) f. ( 3 4) ( ) g. ( ) ( 3) h. (3 3 7) (3 ) i. ( ) (3 ) j. ( ) (3 ) Communication Knowledge/ Understanding 8. For the pairs of polynomials in Question 7, state whether the second is a factor of the first. If not, compare the degree of the remainder to the degree of the divisor. What do you observe? 9. Perform the following divisions: a. ( ) ( 4) b. ( 4 3 ) ( ) c. (4 3 3) ( ) d. ( 5 ) ( ) 0. One factor of is. Find all other factors.. Divide f() by 3.. Divide f() 4 3 by d(). 3. Divide f() by d() 3. 8 CHAPTER

19 Thinking/Inquiry/ Problem Solving 4. In f() d()q() r(), what condition is necessary for d() to be a factor of f()? 5. If f() d()q() r() and r() 0, given that the degree of d() is, what are the possible degrees of r()? Part C Thinking/Inquiry/ Problem Solving 6. If and y are natural numbers and y, then whole numbers q and r must eist such that yq r. a. What is the value of r if y is a factor of? b. If y is not a factor of, what are the possible values of r if y 5, y 7, or y n? 7. a. Divide f() by and write your answer in the form f() ( )q() r. Now divide q() by and write your answer in the form q() ( )Q() r. b. If f () is divided by ( )( ), is Q() in part a the quotient obtained? Justify your answer. c. When f() is divided by ( )( ), can the remainder be epressed in terms of r and r?.3 DIVISION OF POLYNOMIALS 9

20 Section.4 The Remainder Theorem With reference to polynomial functions, we can epress the division algorithm as follows: When a function f() is divided by a divisor d(), producing a quotient q() and a remainder r(), then f() d()q() r(), where the degree of r() is less than the degree of d(). Note that if the divisor is a linear function then the remainder must be a constant. INVESTIGATION The following investigation will illustrate an interesting way in which this relationship can be used.. a. For the function f() 3 7, use long division to divide ( 3 7) by ( ). b. What is the remainder? c. What is the value of f ()?. a. Use long division to divide ( 3 3 ) by ( ). b. What is the remainder? c. What is the value of f()? 3. a. What was the relationship between f() and the remainder in the first division? b. What was the relationship between f() and the remainder in the second division? c. Why do you think we chose the value to use in Question c? d. Why do you think we chose the value to use in Question c? Based on these eamples, complete the following statement: When f() is divided by ( ), then the remainder r() f( ). When f() is divided by ( ), then the remainder r( ) f( ). When f() is divided by ( a), then the remainder r( ) f( ). 0 CHAPTER

21 EXAMPLE Show that for the function f() 3 4, the value of f() is equal to the remainder obtained when f() is divided by ( ). Solution f() () 3 () 4() Since the remainder is 6, then the remainder equals f(). It appears that there is a relationship between the remainder and the value of the function. We now address this in general terms. If the divisor is the linear epression p, we can write the division statement as f() ( p)q() r. This equation is satisfied by all values of. In particular, it is satisfied by p. Replacing with p in the equation we get f(p) (p p)q(p) r (0)q(p) r r. This relationship between the dividend and the remainder is called the Remainder Theorem. The Remainder Theorem If f() is divided by ( p), giving a quotient q() and a remainder r, then r f(p). The Remainder Theorem allows us to determine the remainder in the division of polynomials without performing the actual division, which, as we will see, is a valuable thing to be able to do..4 THE REMAINDER THEOREM

22 EXAMPLE Find the remainder when is divided by a. b. Solution Let f() ; therefore, a. when f() is divided by, the remainder is f(). r f() () 3 4() 5() b. when f() is divided by, the remainder is f(). r f() ( ) 3 4() 5() What do we do if the divisor is not of the form ( p), but of the form (k p)? We have already seen that the remainder in dividing by (k p) is the same as in dividing by p k, so there is no difficulty. In this case, r f p k. EXAMPLE 3 Find the remainder when f() is divided by ( 3). Solution To determine the remainder, we write 3 3 and calculate f 3. The remainder is r f EXAMPLE 4 When 3 3 k 0 is divided by 5, the remainder is 5. Find the value of k. Solution Since r 5 and r f(5), where f(5) k 0, then 0 5k 5 (by the Remainder Theorem) 5k 95 k 39. CHAPTER

23 We have noted that the remainder is always of a degree lower than that of the divisor. In the eamples so far, the divisor was a linear function, so the remainder had to be a constant. In the net eample, the divisor is a quadratic epression, so the remainder can be a linear epression. EXAMPLE 5 Find the remainder when is divided by ( )( ). Solution Using Long Division Epand ( )( ). 4 3 The remainder is Solution Using the Remainder Theorem We have f() Then f() ( )( )q() r() where r() is at most a linear epression. Let r() A B. Now f() ( )( )q() (A B). From f() (3)(0)q() A B A B. From f() 4 () 3 5() 3. So A B. 3 Similarly f() (0)(3)8() (A B) A B and f() () 4 () 3 5() 3 9 so A B 9. 4 We solve equations 3 and 4 for A and B. A B A B 9 Subtracting 3A A 7 and B 5 Since r() A B, the remainder is THE REMAINDER THEOREM 3

24 Eercise.4 Part A Communication. Eplain how you determine the remainder when is divided by.. What is the remainder when is divided by a. b. c. d Determine the remainder in each of the following: a. ( 3) ( 3) b. ( 3 ) ( ) c. ( 3 4 ) ( ) d. (3 4 ) ( ) e. ( 4 5) ( ) f. ( 4 3 ) ( ) Part B Knowledge/ Understanding Application 4. Determine the remainder in each of the following using the Remainder Theorem: a. ( 3 3 4) ( ) b. ( ) ( 3) c. ( 3 3 7) ( ) d. ( 5 ) ( ) e. (6 0 7) (3 ) f. ( ) ( ) g. ( 3 3 ) ( 3) h. ( ) ( ) 5. Determine the value of k in each of the following: a. When 3 k 3 is divided by, the remainder is. b. When 4 k 3 4 is divided by 3, the remainder is 6. c. When 3 3 k is divided by, the remainder is. Thinking/Inquiry/ Problem Solving 6. If f() m 3 g 3 is divided by, the remainder is 3. If f() is divided by, the remainder is 7. What are the values of m and g? 7. If f() m 3 g 3 is divided by, the remainder is 3. If f() is divided by 3, the remainder is. What are the values of m and g? 4 CHAPTER

25 Part C 8. Determine the remainder when ( 3 3 ) is divided by ( 3)( 5). 9. Determine the remainder when ( ) is divided by ( )( ). Thinking/Inquiry/ Problem Solving 0. When is divided into f(), the remainder is 3. Determine the remainder when is divided into each of the following: a. f() b. f() c. f() (4 7) d. f() 7 e. [f()]. If f() ( 5)q() ( 3), what is the first multiple of ( 5) greater than f()?. The epression 4 cannot be factored using known techniques. However, by adding and subtracting, we obtain 4. Therefore, 4 ( ) ( )( ). Use this approach to factor each of the following: a b. 9y 4 8y 4 c d THE REMAINDER THEOREM 5

26 Key Concepts Review After your work in this chapter on Polynomial Functions, you should be familiar with the following concepts: Factoring Types You should be able to identify and simplify epressions of the following types: common trinomial grouping difference of squares Sketching Polynomial Functions Make use of the relationships between -intercepts and the roots of the corresponding equation to sketch the graph of functions. Division of Polynomials Remainder Theorem If f() is divided by ( a), giving a quotient q() and a remainder r, then r f(a). Polynomial Functions from Data The first differences of a linear function are constant. The second differences of a quadratic function are constant. The third differences of a cubic function are constant. 6 CHAPTER

27 CHAPTER : MODELS FOR WATER FLOW RATES investigate and apply wrap-up. Using the data presented in the Career Link, develop and utilize a polynomial mathematical model of the flow-rate and time relationship [Q f(t)] by a. determining the degree of the polynomial, then using the graphing calculator to obtain an algebraic model for Q f(t) with the appropriate polynomial regression function. b. using the graphing calculator to determine the peak flow. When does this occur? Is this a reasonable time for a peak daily flow? Eplain. c. determining an algebraic model for the velocity [V(t)] of the water in the pipe (metres per hour) leaving the water plant if the cross-sectional area [A(t)] of the pipe changes over time with the relationship: A(t) 0.t 0.4 where A(t) is cross-sectional area in square metres, t is time in hours, and Q(t) A(t) V(t). d. verifying that your model in part c is correct using the graphing calculator. Eplain how you did this.. Water travelling at high velocities can cause damage due to ecessive forces at bends (elbows) in pipe networks. If the maimum allowable velocity in this specific pipe is.5 m/s, will the pipe be damaged at the peak flow rate? CAREER LINK WRAP-UP 7

28 Review Eercise. Draw a sketch of each of the following without using your graphing calculator. a. y ( )( 3) b. y ( 3) c. y ( )( 3) d. y ( )( 4)( ) e. y ( ) 3 f. y ( 4)( )( 3) g. y ( ) ( 4) h. y ( ) ( ) i. y ( 3)( ) j. y ( 4)( )( )( 3) k. y ( ) 3 ( 3) l. y ( )( 3). In each of the following, you are given a set of points that lie on the graph of a polynomial function. If possible, determine the equation of the function. a. (, 7), (0, ), (, 5), (, 3), (3, ), (4, 3) b. (0, 4), (, 5), (, 3), (3, 67), (4, 3), (5, 39) c. (, 9), (, 3), (3, 3), (4, 5), (5, 99), (6, 8) d. (, ), (, ), (3, 5), (4, 6) e. (, 75), (, ), (0, ), (, 7), (, 53) 3. Perform the following divisions: a. ( 3 3 ) ( 3) b. ( 3 5 4) ( ) c. (4 3 8 ) ( ) d. ( ) ( ) 4. Without using long division, determine the remainder when a. ( ) is divided by ( ). b. ( 3 4 ) is divided by ( ). c. ( 3 5 ) is divided by ( ). d. ( 4 3 3) is divided by ( ). e. (3 3 ) is divided by (3 ). 8 CHAPTER

29 5. Divide each polynomial by the factor given, then epress each polynomial in factored form. a. 3, given is a factor. b , given 3 is a factor. c , given 3 is a factor. 6. a. When 3 3k 5 is divided by, the remainder is 9. Find the value of k. b. When r 3 g 4 is divided by, the remainder is. When it is divided by 3, the remainder is 0. Find the values of r and g. REVIEW EXERCISE 9

30 Chapter Test Achievement Category Knowledge/Understanding Questions, 3, 5, 7b Thinking/Inquiry/Problem Solving 8 Communication 4 Application, 6, 7a, 9. Factor each of the following: a. 8 50y b. pm 3 m pm c. 6 d. 6y y 9. Without using a graphing calculator, sketch the graph of a. y ( )( )( 3) b. y ( ) 3. Find the quotient and remainder when a is divided by. b. ( 3 6 ) is divided by ( 3). 4. Since f() 0 for f() 4 3 6, do you think ( ) is a factor of f() 4 3 6? Eplain. 5. Without using long division, find the remainder when ( ) is divided by ( ). 6. Find the value of k if there is a remainder of 7 when k is divided by ( ). 7. a. Do (, ), (, ), (3, ), (4, 5) lie on the graph of a quadratic function? b. Use your graphing calculator to find the simplest polynomial function that contains the following points: (, 4), (, 6), (3, 34), (4, 9). 8. When 3 c d is divided by, the remainder is 3, and when it is divided by, the remainder is 3. Determine the values of c and d. 9. One factor of is. Determine the other factors. 30 CHAPTER

31 Chapter POLYNOMIAL EQUATIONS AND INEQUALITIES It s happened to everyone. You ve lost your favourite CD, and your room is an unbelievable mess. Rather than attempting to sort through everything, why not consider a few key places where it could be, and eamine these areas closely until you find your CD. Similarly, if a manufacturer discovers a flaw in her product, the key intermediate assembly stages are eamined individually until the source of the problem is found. These are two eamples of a general learning and problem solving strategy: consider a thing in terms of its component parts, without losing sight of the fact that the parts go together. This problem solving strategy is a great way to solve mathematical equations, as well. In this chapter, you will see that polynomial equations can be solved using the same strategy you might use for finding a lost CD. Just eamine the key component factors until you solve the problem! CHAPTER EXPECTATIONS In this chapter, you will understand the Remainder and Factor Theorems, Section. factor polynomial epressions, Section. compare the nature of change in polynomial functions with that of linear and quadratic functions, Section.3 determine the roots of polynomial equations, Section.3 determine the real roots of non-factorable polynomial equations, Section.4 solve problems involving the abstract etensions of algorithms, Section.4 solve factorable and non-factorable polynomial inequalities, Section.5 write the equation of a family of polynomial functions, Section.5 write the equation of a family of polynomial functions, Career Link describe intervals and distances, Section.6

32 Review of Prerequisite Skills To begin your study of Polynomial Equations and Inequalities in Chapter, you should be familiar with the following skills: Solving Linear Equations and Inequalities 4( 3) Evaluating Polynomial Functions If f () 3 3 7, then f () () 3 3() 7 7. Factoring Quadratic Polynomials 7 ( 3)( 4) ( 7)(3 ) Solving Quadratic Equations by Factoring (3 )( ) 0 (3 )(4 5) or or or 3 or 5 4 Solving Quadratic Equations Using the Quadratic Formula For the equation a b bac b c 0, 4 a. For 6 0, a 6, For 6 3 0, a 6, b, c. b, c 3. 6)() 4( 4 6)(3) 4( or or 0.9 (correct to one decimal) 3 CHAPTER

33 Eercise. Solve. a. 3 5 b. 3( ) 7 3( 7) c. 7 ( 3) 9 5 d. ( 3)( ) 5. Solve and graph on the real number line. a. 3 5 b c. 4 5 ( 7) d Evaluate each of the following for f() 3. a. f () b. f () c. f (3) d. f 4. Evaluate each of the following for f() a. f () b. f () c. f (3) d. f 5. Factor each of the following fully. a b. y 3y c d e f g. 4 0 h. 3 3 i Solve. a. ( 4) 0 b. ( 3)( ) 0 c d. y 9y 8 0 e. 5 0 f g h i Recall that the quadratic formula to solve the quadratic equation a b bac b c 0, C is 4 a. Solve correct to one decimal place. a b. 3y 5y 4 0 c d e. 3 5 f. 6y 5y 6 g. p 3p 5 0 h i. ( 5)( )( 3) REVIEW OF PREREQUISITE SKILLS 33

34 investigate CHAPTER : RESEARCHING DOSE RESPONSE RELATIONSHIPS In response to the health concerns of Canada s aging population, the pharmaceutical industry has dramatically increased its investment in research over the past ten years. A key component of the research process is the generation of mathematical models that predict dose-response relationships. Dose refers to the quantity of medication administered to a patient, and response refers to the effect on the patient. For eample, the dose-response relationship for asthma medication may be in terms of the mass of drug administered versus the percentage increase in lung capacity. Polynomial equations are often used to model the dose-response relationship because they can be fit to a data set that changes slope a number of times and may cross the -ais multiple times (i.e., it may feature multiple roots). In this chapter, you will develop the algebraic tools to solve polynomial equations and inequalities, then you will investigate the properties of polynomial roots and the absolute-value function. Case Study Pharmaceutical Researcher Pharmaceutical companies are, of course, also interested in modelling the side-effect responses of medication. For eample, the equation R(t) 5t 4 4t 3 9 t 4 3 t can be used to model the side-effect response [R(t)] in degrees Celsius above or below the normal body temperature (36.9 C) of an eperimental drug t hours after it was administered. The equation is valid for 0 t. hours. Due to the stress of temperature change on the body, a second drug is administered at the moment the patient s temperature starts to eceed 36.9 C. DISCUSSION QUESTIONS. Within the contet of the problem, what happens when R(t) crosses the t-ais?. Using your prior knowledge of linear and quadratic functions and your work in Chapter, predict how many times the second drug will have to be administered. (Hint: Think about the degree of the function.) Eplain using a rough sketch. Do not make a table of values or plot the graph. 3. Once again using your prior knowledge of linear and quadratic functions and your work in Chapter, predict how many times the patient s temperature can spike (i.e., reach a maimum or minimum). Eplain using a rough sketch. At the end of this chapter you will apply the tools of solving polynomial equations and inequalities in assessing the performance of the eperimental drug introduced above. 34 CHAPTER

35 Section. The Factor Theorem The Remainder Theorem tells us that when we divide 5 6 by 3, the remainder is f(3) (3) 5(3) Since the remainder is zero, 5 6 is divisible by ( 3). By divisible, we mean evenly divisible. If f() is divisible by p, we say p is a factor of f(). On the other hand, if we divide 5 6 by ( ), the remainder is f() () 5() 6. The fact that the remainder is not zero tells us that 5 6 is not evenly divisible by ( ). That is, ( ) is not a factor of 5 6. The Remainder Theorem tells us that if the remainder is zero on division by ( p), then f (p) 0. If the remainder is zero, then ( p) divides evenly into f(), and ( p) is a factor of f(). Conversely, if p is a factor of f(), then the remainder f(p) must equal zero. These two statements give us the Factor Theorem, which is an etension of the Remainder Theorem. The Factor Theorem ( p) is a factor of f() if and only if f(p) 0. EXAMPLE Show that is a factor of Solution f () 3 3() 5() 6 0 Since f () 0, is a factor of Solution 3 Dividing THE FACTOR THEOREM 35

36 Since the remainder is zero, is a factor of Both solutions verify that is a factor. Note that Solution tells us that the second factor is 3. EXAMPLE Is ( ) a factor of f() ? Solution We test if is a factor by evaluating f(). f () () 3 3() 5() Since f () 0, ( ) is not a factor of In using the Factor Theorem, we must find a value p so that f(p) 0. We can then say that ( p) is a factor, and by division we can determine the second factor. The question is how to determine the value of p. This is partly a matter of guessing, but we make the guessing easier by noting that there is a limited number of possible values. In the function f() , if there is a value for p such that f(p) 0, then p 3 p 3p 6 0. We are, of course, interested in integer values for p. Note that p 3 p 3p 6 means that p(p p 3) 6. If p is an integer, then p p 3 is an integer, so the only possible values for p are,, 3, 6, and we need to consider only these. In other words, the only possible values for p are divisors of the constant term in f(). EXAMPLE 3 Factor Solution Possible values for p are,, 3, 4, 6,, and 4. f() f() f() Therefore ( ) is a factor of f(). 36 CHAPTER To find the other factor(s), one method is to use long division and divide by as follows:

37 Factoring further, ( 4)( 3). Therefore, ( )( 4)( 3). An alternative is to use the following method of comparing coefficients: ( )( k ) 3 k k 4 3 (k ) ( k) 4 Comparing the coefficients of, k, so k. A check can be obtained by comparing the coefficients of, which gives k 4 and k. Therefore, ( )( ), and further factoring gives ( )( 3)( 4). EXAMPLE 4 Factor f() t chnology e Solution Possible values of p such that f(p) 0 are,, 3, 6, 9, and 8. Checking all of these is time-consuming. We can help ourselves by using a calculator to sketch the graph y f(). From the graph there are three potential integer values: p, p, and p 8. But p 8 is impossible, since it is not a divisor of 8. Now f() f() () 3 9() 5() 8 0. Therefore is a factor of f (). To find the other factor(s), you can use long division or you can compare coefficients Since 7 9 cannot be factored further, ( )( 7 9). EXAMPLE 5 Factor f() THE FACTOR THEOREM 37

38 t chnology e Solution Possible values of p such that f(p) 0 are,, 3, 4, 6, and. From the graph, values to check are p 6, p, and p. Note that the fourth -intercept is between and 3 and is not an integer. Graph y f () and use the :value mode under the CALCULATE menu or substitute to evaluate f (6), f (), and f(). f(6) 6 4 3(6) 3 3(6) 3(6) 0 f() f() Therefore and are factors of f (). To determine the other factor(s), use the method of comparing coefficients ( )( )( k ) ( )( k ) 4 k 3 k 4 k 3 k Since k 3 3 3, k 3. Now ( )( 3 ). All eamples considered here involve monic polynomials. A monic polynomial has one as the coefficient of its highest degree term. In the net section, we will consider the use of the Factor Theorem with polynomials having first-term coefficients other than one. EXAMPLE 6 Factor 3 y 3. Solution Consider this as a function of. That is, f() 3 y 3. Since f(y) y 3 y 3 0, then, by the Factor Theorem, ( y) is a factor of 3 y 3. Divide to obtain the other factor(s). y y y 3 0 y 0y y 3 3 y y 0y y y y y 3 y y 3 0 Therefore 3 y 3 ( y)( y y ). 38 CHAPTER

39 The epression 3 y 3 is referred to as a difference of cubes and it occurs often enough that its factorization is worth memorizing: 3 y 3 ( y)( y y ). Since 3 y 3 3 (y) 3, we have 3 y 3 ( (y))( (y) (y) ( y)( y y ). The epression 3 y 3 is referred to as a sum of cubes. By the same process as above, 3 y 3 ( y)( y y ). The Sum and Difference of Cubes 3 y 3 ( y)( y y ) 3 y 3 ( y)( y y ) EXAMPLE 7 Factor 7u Solution Since 7u 3 64 (3u) 3 (4) 3 (3u 4)((3u) (3u)(4) (4) ) (3u 4)(9u u 6). Eercise. Part A. If ( 8) is a factor of f(), then what is the value of f(8)? Communication. a. If f(5) 0, state a factor of f(). b. Eplain how you would find the other factors of f(). 3. If f() is equal to 0 when or or 3, what are the factors of f()? Eplain. 4. In each of the following, determine whether the binomial is a factor of f(). a. ; f() 7 6 b. ; f() 8 6 c. ; f() d. 3; f() e. ; f() f. ; f() THE FACTOR THEOREM 39

40 Part B Knowledge/ Understanding 5. If f() 3 3, a. show that f (3) 0. b. what is a linear factor of f ()? c. find the quadratic factor by long division. 6. If g() 3 5 6, a. show that g() 0. b. what is a linear factor of g()? c. find the quadratic factor by the method of comparing coefficients. Application Thinking/Inquiry/ Problem Solving Knowledge/ Understanding 7. Completely factor the following: a b. 3 c. y 3 9y 9y d e. y 3 y y f g h If ( ) is a factor of 3 k 3, what is the value of k? 9. If 3 4 k 5 is divisible by ( ), what is the value of k? 0. Using the formulas for factoring the sum or difference of cubes, factor each of the following: a. 3 7 b. y 3 8 c. 5u 3 64r 3 d. 000w 3 y 3 e. ( y) 3 u 3 z 3 f. 5u 3 40( y) 3 Part C. Use the Factor Theorem to prove that is divisible by. Thinking/Inquiry/ Problem Solving. a. Show that y is a factor of 4 y 4. b. What is the other factor? c. Factor a. Show that y is a factor of 5 y 5. b. What is the other factor? c. Factor CHAPTER

41 4. a. Show that y is a factor of n y n. b. What is the other factor? 5. Prove that ( a) is a factor of ( a) 5 ( c) 5 (a c) Prove that ( a) is a factor of 3 (a b c) (ab bc ca) abc. 7. If n N, under what conditions will ( y) be a factor of n y n? 8. Factor 5 y Does the epression 3 5 have any rational factors? Eplain.. THE FACTOR THEOREM 4

42 Section. The Factor Theorem Etended We have seen that if an epression such as has a factor ( k), where k is an integer, then k must be a divisor of 6. In order to determine which, if any, of the divisors of 6 could be a value for k, we used a graphing calculator to determine a suitable value. In this eample, k 3 and ( 3)( ). What happens when the coefficient of the highest-order term is an integer other than? Let s consider an eample. EXAMPLE Factor f () Solution If the factors have integers as coefficients, then the first terms must be 3,, and, and the second terms must be 7,, and, with the signs in the factors to be determined. We might have a factor such as (3 ) or (3 7) or ( 7). Are there other possibilities? t chnology e APPENDIX P. 48 We draw the graph and search for possible values. The graph shown indicates three intercepts, but it is not easy to determine their values. Restricting the domain to X min and X ma 5 shows more clearly that there are three possible values for k. They are approimately k 3, k 5 3, and k 3 3, since the only denominator we can use is 3 (because 3 and are the only divisors of the coefficient of 3 ). Of these, only 3 is a possible value for k, because the numerator must be a divisor of 7. Since f 3 0, 3 is a factor. By long division, or the method of comparing coefficients, (3 )( 6 7). Since 6 7 has no integer factors, we are done. A function f () a n n a n n a 0 has a factor (q p), if f p q 0, where q divides a n and p divides a 0. 4 CHAPTER

43 EXAMPLE Factor f() t chnology e Solution A graph of the function shows that only one value of k p q is possible, that it lies between and, and that it is close to. Since p is a divisor of 0 and q is a divisor of 6, a good guess for k is 5 3. Evaluating, f 5 3 0, so (3 5)( 3 ). EXAMPLE 3 Factor fully f() t chnology e Solution A graph of the function shows that k p q can be between and 0, or it can be between and 3. Since p divides 0 and q divides 6, we try k 3 and k 5, obtaining f 3 0 and f 5 0. Therefore, two factors are (3 ) and ( 5). The other factors can be determined by division or by comparison. Then (3 )( 5)( ), and the third factor cannot be simplified in integers. Eercise. Communication Part A. For each of the following, eplain how you could find the values p of that q potentially could make the polynomial have a value of zero. State all the possible values of p q. a. 9 5 b c d e f THE FACTOR THEOREM EXTENDED 43

44 Part B Application. A cubic function f () with integral coefficients has the following properties: f 3 0, ( ) is a factor of f (), and f (4) 50. Determine f (). 3. A cubic function g() with integral coefficients has the following properties: g(3) 0, g 3 4 0, ( ) is a factor of g(), g() 84. Determine g(). Knowledge/ Understanding 4. Factor fully the following polynomials: a. 3 b c d e f g h Part C 5. Factor fully the following epressions: a. p 3 (p q) (p q) q b. ab 3 (a b ab) (b a ) 44 CHAPTER

45 Section.3 Solving Polynomial Equations In earlier grades, you learned to solve linear and quadratic equations. A quadratic equation can be solved by factoring (if possible) or by the quadratic formula b bac 4 a. We now consider solutions to higher-order equations. There are formulas for solving a general cubic equation and a general quartic equation, but they are quite complicated. The following eamples demonstrate strategies that you can use in solving cubic, quartic, and other higher-order equations. Any equation of the form f() 0 can be solved if f() can be epressed as a combination of linear and quadratic factors. The first strategy, then, is to factor f(). It may be possible to factor f() by familiar methods, such as grouping terms. If not, we can employ the Factor Theorem. We will assume that unless otherwise stated, we are to solve all equations using the set of comple numbers, C, as the domain. In general, if the domain is C, a polynomial equation of degree n has n roots. EXAMPLE Solve t chnology e Solution The pattern of coefficients (,, 9, 9) suggests grouping the terms ( ) 9( ) 0 ( )( 9) 0 ( )( 3)( 3) 0 Then 0 or 3 0 or 3 0 so or 3 or 3. Note that, 3, and 3 are the -intercepts of the graph of the corresponding cubic function y EXAMPLE Solve Solution Since there is no obvious way of grouping, the Factor Theorem is employed. The factors of 5 are, and 5. If f() 3 4 5, then f() 4 5 0, therefore ( ) is a factor..3 SOLVING POLYNOMIAL EQUATIONS 45

46 There are two methods we can use for finding the second factor: Comparing Coefficients ( )( k 5) 3 k 5 k 5 3 (k ) (5 k) 5 Therefore k 0 or 5 k 4 k k Using Long Division The second factor is The second factor is 5 = 0. 5 = 0. Then ( )( 5) The equation becomes ( )( 5) 0 Then 0 or or 9i EXAMPLE 3 Solve t chnology e Solution The graph of f() is shown. If there are integer roots, they must be either or 5. From the graph, one possible root is. Checking, f() 0. Then, by long division or the method of comparison, f() ( )( 8 5). Therefore, and ( )( 8 5) 0 or The solutions are, CHAPTER

47 EXAMPLE 4 Solve t chnology e Solution Let f () Since f() is not a monic polynomial, non-integer rational roots are possible for f() 0. Since p is a divisor of 6 and q is a divisor of, the possible values for q p are,,, 3, 6, and 3. Graphing f() on a graphing calculator using X min and X ma as our domain, we find that one root of f() 0 lies in the interval.34.9, another lies in the interval.46.5, and the third root appears to be. Checking our list of values for q p, we see that the only possible values are 3,, and. Checking, we find that f 3 f f() 0. Therefore f() (3 )( )( ) and the roots of f () 0 are 3,, and. EXAMPLE 5 a. Find the family of cubic functions whose -intercepts are,, and 3. b. Find the particular member of the above family whose graph passes through the point (, 0). Solution a. Since,, and 3 are -intercepts, ( ), ( ), and ( 3) must be factors of the cubic function. Therefore, y k( )( )( 3), where k is a constant, represents the family of cubic functions. b. If (, 0) lies on the graph of one member of the above family, then (, 0) must satisfy its equation. Substituting, we get 0 k(4)()() or k 5. Therefore, the particular cubic function is y 5( )( )( 3). We frequently encounter equations that cannot be factored. In such situations, the best we can hope for is to determine approimate values for the roots using a graphing calculator. The following eample illustrates this application. EXAMPLE 6 Solve Solution Can the epression be factored?.3 SOLVING POLYNOMIAL EQUATIONS 47

48 The only factors of are and, and none is a root of the equation. t chnology e Using a graphing calculator, graph f() 3 5. Using the ZDecimal instruction in the ZOOM mode, note from the graph that there is a root slightly to the left of. CALC TRACE Press nd for the CALCULATE menu. Select :zero and press ENTER. Input as the Left Bound. Input as the Right Bound. Input as the Guess. The approimate root is Referring back to the graph, you will notice that there is another root close to. Use and 3 as the left and right bounds and as your guess. The second root is approimately Use a similar procedure for the other root between 4 and 5. This third root is approimately The roots of the equation are.3,.9, and 4.4, to one decimal place. Note that this procedure will find approimate real roots but will not find the non-real roots (if there are any) of a polynomial equation. EXAMPLE 7 Solve: Solution ( 5)( ) 0 5 or Then 5 or i. Sometimes making a substitution assists in the solving of higher-order equations. In Eample 7, you could let y to get the equation y 4y 5. Solve for y and then solve for. In the net eample, the substitution is more subtle. 48 CHAPTER