Algebra and Trigonometry

James Stewart, Lothar Redlin, Saleem Watson

Chapter 4

Polynomial and Rational Functions - all with Video Answers

Educators

Section 1

Polynomial Functions and Their Graphs

Problem 1

Sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^{n}$ from Figure $2 .$ Indicate all $x-$ and $y$ -intercepts on each graph.
$$\begin{array}{ll}{\text { (a) } P(x)=x^{2}-4} & {\text { (b) } Q(x)=(x-4)^{2}} \\ {\text { (c) } R(x)=2 x^{2}-2} & {\text { (d) } S(x)=2(x-2)^{2}}\end{array}$$

Peter W.

Peter W.

Numerade Educator

Problem 2

Sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^{n}$ from Figure $2 .$ Indicate all $x-$ and $y$ -intercepts on each graph.
$$
\begin{array}{l}{\text { (a) } P(x)=x^{4}-16 \quad \text { (b) } Q(x)=(x+2)^{4}} \\ {\text { (c) } R(x)=(x+2)^{4}-16 \text { (d) } S(x)=-2(x+2)^{4}}\end{array}
$$

Peter W.

Numerade Educator

Problem 3

Sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^{n}$ from Figure $2 .$ Indicate all $x-$ and $y$ -intercepts on each graph.
$$
\begin{array}{ll}{\text { (a) } P(x)=x^{3}-8} & {\text { (b) } Q(x)=-x^{3}+27} \\ {\text { (c) } R(x)=-(x+2)^{3}} & {\text { (d) } S(x)=\frac{1}{2}(x-1)^{3}+4}\end{array}
$$

Peter W.

Numerade Educator

Problem 4

Sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^{n}$ from Figure $2 .$ Indicate all $x-$ and $y$ -intercepts on each graph.
$$
\begin{array}{ll}{\text { (a) } P(x)=(x+3)^{5}} & {\text { (b) } Q(x)=2(x+3)^{5}-64} \\ {\text { (c) } R(x)=-\frac{1}{2}(x-2)^{5}} & {\text { (d) } S(x)=-\frac{1}{2}(x-2)^{5}+16}\end{array}
$$

Peter W.

Numerade Educator

Problem 5

Match the polynomial function with one of the graphs I-VI. Give reasons for your choice.
$$
P(x)=x\left(x^{2}-4\right)
$$

Peter W.

Numerade Educator

Problem 6

Match the polynomial function with one of the graphs I-VI. Give reasons for your choice.
$$
Q(x)=-x^{2}\left(x^{2}-4\right)
$$

Peter W.

Numerade Educator

Problem 7

Match the polynomial function with one of the graphs I-VI. Give reasons for your choice.
$$
R(x)=-x^{5}+5 x^{3}-4 x
$$

Peter W.

Numerade Educator

Problem 8

Match the polynomial function with one of the graphs I-VI. Give reasons for your choice.
$$
S(x)=\frac{1}{2} x^{6}-2 x^{4}
$$

Peter W.

Numerade Educator

Problem 9

Match the polynomial function with one of the graphs I-VI. Give reasons for your choice.
$$
T(x)=x^{4}+2 x^{3}
$$

Peter W.

Numerade Educator

Problem 10

Match the polynomial function with one of the graphs I-VI. Give reasons for your choice.
$$
U(x)=-x^{3}+2 x^{2}
$$

Peter W.

Numerade Educator

Problem 11

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=(x-1)(x+2)
$$

AG

Ankit G.

Numerade Educator

Problem 12

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=(x-1)(x+1)(x-2)
$$

AG

Ankit G.

Numerade Educator

Problem 13

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=x(x-3)(x+2)
$$

AG

Ankit G.

Numerade Educator

Problem 14

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=(2 x-1)(x+1)(x+3)
$$

AG

Ankit G.

Numerade Educator

Problem 15

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=(x-3)(x+2)(3 x-2)
$$

AG

Ankit G.

Numerade Educator

Problem 16

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=\frac{1}{5} x(x-5)^{2}
$$

AG

Ankit G.

Numerade Educator

Problem 17

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=(x-1)^{2}(x-3)
$$

Peter W.

Numerade Educator

Problem 18

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=\frac{1}{4}(x+1)^{3}(x-3)
$$

Peter W.

Numerade Educator

Problem 19

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=\frac{1}{12}(x+2)^{2}(x-3)^{2}
$$

AG

Ankit G.

Numerade Educator

Problem 20

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=(x-1)^{2}(x+2)^{3}
$$

AG

Ankit G.

Numerade Educator

Problem 21

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=x^{3}(x+2)(x-3)^{2}
$$

AG

Ankit G.

Numerade Educator

Problem 22

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.
$$
P(x)=(x-3)^{2}(x+1)^{2}
$$

AG

Ankit G.

Numerade Educator

Problem 23

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=x^{3}-x^{2}-6 x
$$

AG

Ankit G.

Numerade Educator

Problem 24

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=x^{3}+2 x^{2}-8 x
$$

AG

Ankit G.

Numerade Educator

Problem 25

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=-x^{3}+x^{2}+12 x
$$

AG

Ankit G.

Numerade Educator

Problem 26

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=-2 x^{3}-x^{2}+x
$$

AG

Ankit G.

Numerade Educator

Problem 27

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=x^{4}-3 x^{3}+2 x^{2}
$$

AG

Ankit G.

Numerade Educator

Problem 28

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=x^{5}-9 x^{3}
$$

AG

Ankit G.

Numerade Educator

Problem 29

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=x^{3}+x^{2}-x-1
$$

AG

Ankit G.

Numerade Educator

Problem 30

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=x^{3}+3 x^{2}-4 x-12
$$

AG

Ankit G.

Numerade Educator

Problem 31

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=2 x^{3}-x^{2}-18 x+9
$$

AG

Ankit G.

Numerade Educator

Problem 32

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=\frac{1}{8}\left(2 x^{4}+3 x^{3}-16 x-24\right)^{2}
$$

AG

Ankit G.

Numerade Educator

Problem 33

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=x^{4}-2 x^{3}-8 x+16
$$

Check back soon!

Problem 34

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=x^{4}-2 x^{3}+8 x-16
$$

Check back soon!

Problem 35

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=x^{4}-3 x^{2}-4
$$

AG

Ankit G.

Numerade Educator

Problem 36

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph.
$$
P(x)=x^{6}-2 x^{3}+1
$$

AG

Ankit G.

Numerade Educator

Problem 37

Determine the end behavior of $P$ . Compare the graphs of $P$ and $Q$ on large and small viewing rectangles, as in Example 3$(b)$ .
$$P(x)=3 x^{3}-x^{2}+5 x+1 ; \quad Q(x)=3 x^{3}$$

AG

Ankit G.

Numerade Educator

Problem 38

Determine the end behavior of $P$ . Compare the graphs of $P$ and $Q$ on large and small viewing rectangles, as in Example 3$(b)$ .
$$
P(x)=-\frac{1}{8} x^{3}+\frac{1}{4} x^{2}+12 x ; \quad Q(x)=-\frac{1}{8} x^{3}
$$

AG

Ankit G.

Numerade Educator

Problem 39

Determine the end behavior of $P$ . Compare the graphs of $P$ and $Q$ on large and small viewing rectangles, as in Example 3$(b)$ .
$$
P(x)=x^{4}-7 x^{2}+5 x+5 ; \quad Q(x)=x^{4}
$$

AG

Ankit G.

Numerade Educator

Problem 40

Determine the end behavior of $P$ . Compare the graphs of $P$ and $Q$ on large and small viewing rectangles, as in Example 3$(b)$ .
$$
P(x)=-x^{5}+2 x^{2}+x ; \quad Q(x)=-x^{5}
$$

AG

Ankit G.

Numerade Educator

Problem 41

Determine the end behavior of $P$ . Compare the graphs of $P$ and $Q$ on large and small viewing rectangles, as in Example 3$(b)$ .
$$
P(x)=x^{11}-9 x^{9} ; \quad Q(x)=x^{11}
$$

AG

Ankit G.

Numerade Educator

Problem 42

Determine the end behavior of $P$ . Compare the graphs of $P$ and $Q$ on large and small viewing rectangles, as in Example 3$(b)$ .
$$
P(x)=2 x^{2}-x^{12} ; \quad Q(x)=-x^{12}
$$

AG

Ankit G.

Numerade Educator

Problem 43

The graph of a polynomial function is given. From the graph, find
(a) the $x$ - and $y$ -intercepts
(b) the coordinates of all local extrema
$$
P(x)=-x^{2}+4 x
$$

Aamir M.

Numerade Educator

Problem 44

The graph of a polynomial function is given. From the graph, find
(a) the $x$ - and $y$ -intercepts
(b) the coordinates of all local extrema
$$
P(x)=\frac{2}{9} x^{3}-x^{2}
$$

Aamir M.

Numerade Educator

Problem 45

The graph of a polynomial function is given. From the graph, find
(a) the $x$ - and $y$ -intercepts
(b) the coordinates of all local extrema
$$
P(x)=-\frac{1}{2} x^{3}+\frac{3}{2} x-1
$$

AG

Ankit G.

Numerade Educator

Problem 46

The graph of a polynomial function is given. From the graph, find
(a) the $x$ - and $y$ -intercepts
(b) the coordinates of all local extrema
$$
P(x)=\frac{1}{9} x^{4}-\frac{4}{9} x^{3}
$$

AG

Ankit G.

Numerade Educator

Problem 47

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$
y=-x^{2}+8 x, \quad[-4,12] \text { by }[-50,30]
$$

Peter W.

Numerade Educator

Problem 48

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$
y=x^{3}-3 x^{2}, \quad[-2,5] \text { by }[-10,10]
$$

Peter W.

Numerade Educator

Problem 49

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$
y=x^{3}-12 x+9, \quad[-5,5] \text { by }[-30,30]
$$

Peter W.

Numerade Educator

Problem 50

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$
y=2 x^{3}-3 x^{2}-12 x-32,[-5,5] \text { by }[-60,30]
$$

Peter W.

Numerade Educator

Problem 51

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$
y=x^{4}+4 x^{3}, \quad[-5,5] \text { by }[-30,30]
$$

Peter W.

Numerade Educator

Problem 52

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$
y=x^{4}-18 x^{2}+32, \quad[-5,5] \text { by }[-100,100]
$$

Peter W.

Numerade Educator

Problem 53

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$
y=3 x^{5}-5 x^{3}+3, \quad[-3,3] \text { by }[-5,10]
$$

Peter W.

Numerade Educator

Problem 54

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$
y=x^{5}-5 x^{2}+6, \quad[-3,3] \text { by }[-5,10]
$$

Peter W.

Numerade Educator

Problem 55

Graph the polynomial and determine how many local maxima and minima it has.
$$
y=-2 x^{2}+3 x+5
$$

AG

Ankit G.

Numerade Educator

Problem 56

Graph the polynomial and determine how many local maxima and minima it has.
$$
y=x^{3}+12 x
$$

AG

Ankit G.

Numerade Educator

Problem 57

Graph the polynomial and determine how many local maxima and minima it has.
$$
y=x^{3}-x^{2}-x
$$

AG

Ankit G.

Numerade Educator

Problem 58

Graph the polynomial and determine how many local maxima and minima it has.
$$
y=6 x^{3}+3 x+1
$$

AG

Ankit G.

Numerade Educator

Problem 59

Graph the polynomial and determine how many local maxima and minima it has.
$$
y=x^{4}-5 x^{2}+4
$$

AG

Ankit G.

Numerade Educator

Problem 60

Graph the polynomial and determine how many local maxima and minima it has.
$$
y=1.2 x^{5}+3.75 x^{4}-7 x^{3}-15 x^{2}+18 x
$$

AG

Ankit G.

Numerade Educator

Problem 61

Graph the polynomial and determine how many local maxima and minima it has.
$$
y=(x-2)^{5}+32
$$

AG

Ankit G.

Numerade Educator

Problem 62

Graph the polynomial and determine how many local maxima and minima it has.
$$
y=\left(x^{2}-2\right)^{3}
$$

AG

Ankit G.

Numerade Educator

Problem 63

Graph the polynomial and determine how many local maxima and minima it has.
$$
y=x^{8}-3 x^{4}+x
$$

AG

Ankit G.

Numerade Educator

Problem 64

Graph the polynomial and determine how many local maxima and minima it has.
$$
y=\frac{1}{3} x^{7}-17 x^{2}+7
$$

AG

Ankit G.

Numerade Educator

Problem 65

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$
P(x)=c x^{3} ; \quad c=1,2,5, \frac{1}{2}
$$

AG

Ankit G.

Numerade Educator

Problem 66

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$
P(x)=(x-c)^{4} ; \quad c=-1,0,1,2
$$

AG

Ankit G.

Numerade Educator

Problem 67

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$
P(x)=x^{4}+c ; \quad c=-1,0,1,2
$$

AG

Ankit G.

Numerade Educator

Problem 68

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$
P(x)=x^{3}+c x ; \quad c=2,0,-2,-4
$$

AG

Ankit G.

Numerade Educator

Problem 69

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$
P(x)=x^{4}-c x ; \quad c=0,1,8,27
$$

AG

Ankit G.

Numerade Educator

Problem 70

Graph the family of polynomials in the same viewing rectangle, using the given values of $c .$ Explain how changing the value of $c$ affects the graph.
$$
P(x)=x^{c} ; \quad c=1,3,5,7
$$

AG

Ankit G.

Numerade Educator

Problem 71

(a) On the same coordinate axes, sketch graphs (as accu-
rately as possible) of the functions
$y=x^{3}-2 x^{2}-x+2 \quad$ and $\quad y=-x^{2}+5 x+2$
(b) Based on your sketch in part (a), at how many points do
the two graphs appear to intersect?
(c) Find the coordinates of all intersection points.

Peter W.

Numerade Educator

Problem 72

Portions of the graphs of $y=x^{2}, y=x^{3}, y=x^{4}, y=x^{5},$ and $y=x^{6}$ are plotted in the figures. Determine which function belongs to each graph.

AG

Ankit G.

Numerade Educator

Problem 73

Recall that a function $f$ is odd if $f(-x)=-f(x)$ or even if $f(-x)=f(x)$ for all real $x .$
$$\begin{array}{l}{\text { (a) Show that a polynomial } P(x) \text { that contains only odd }} \\ {\text { powers of } x \text { is an odd function. }} \\ {\text { (b) Show that a polynomial } P(x) \text { that contains only even }} \\ {\text { powers of } x \text { is an even function. }} \\ {\text { (c) Show that if a polynomial } P(x) \text { contains both odd and }} \\ {\text { even powers of } x, \text { then it is neither an odd nor an even }} \\ {\text { function. }}\end{array}$$
$$
\begin{array}{l}{\text { (d) Express the function }} \\ {\qquad P(x)=x^{5}+6 x^{3}-x^{2}-2 x+5} \\ {\text { as the sum of an odd function and an even function. }}\end{array}
$$

Peter W.

Numerade Educator

Problem 74

(a) Graph the function $P(x)=(x-1)(x-3)(x-4)$ and find all local extrema, correct to the nearest tenth.
(b) Graph the function
$$Q(x)=(x-1)(x-3)(x-4)+5$$
and use your answers to part (a) to find all local extrema, correct to the nearest tenth.

Peter W.

Numerade Educator

Problem 75

(a) Graph the function $P(x)=(x-2)(x-4)(x-5)$ and determine how many local extrema it has.
(b) If $a<b<c,$ explain why the function
$$P(x)=(x-a)(x-b)(x-c)$$
must have two local extrema.

Peter W.

Numerade Educator

Problem 76

(a) How many $x$ -intercepts and how many local extrema
does the polynomial $P(x)=x^{3}-4 x$ have?
(b) How many $x$ -intercepts and how many local extrema
does the polynomial $Q(x)=x^{3}+4 x$ have?
(c) If $a>0,$ how many $Q(x)=x^{3}+4 x$ how many local
extrema does each of the polynomials $P(x)=x^{3}-a x$
and $Q(x)=x^{3}+a x$ have? Explain your answer.

Peter W.

Numerade Educator

Problem 77

Market Research $\quad$ A market analyst working for a small-appliance manufacturer finds that if the firm produces and sells $x$ blenders annually, the total profit (in dollars) is $$P(x)=8 x+0.3 x^{2}-0.0013 x^{3}-372$$
Graph the function $P$ in an appropriate viewing rectangle and use the graph to answer the following questions.
(a) When just a few blenders are manufactured, the firm loses money (profit is negative). (For example, $P(10)=-263.3,$ so the firm loses $\$ 263.30$ if it produces and sells only 10 blenders.) How many blenders must the firm produce to break even?
(b) Does profit increase indefinitely as more blenders are produced and sold? If not, what is the largest possible profit the firm could have?

Sheryl E.

Numerade Educator

Problem 78

Population Change The rabbit population on a small is- land is observed to be given by the function $$P(t)=120 t-0.4 t^{4}+1000$$ where $t$ is the time (in months) since observations of the island began.
(a) When is the maximum population attained, and what is that maximum population?
(b) When does the rabbit population disappear from the island?

Peter W.

Numerade Educator

Problem 79

Volume of a Box An open box is to be constructed from a piece of cardboard 20 cm by 40 cm by cutting squares of side length x from each corner and folding up the sides, as
shown in the figure.
$$
\begin{array}{l}{\text { (a) Express the volume } V \text { of the box as a function of } x .} \\ {\text { (b) What is the domain of } V ? \text { (Use the fact that length and }} \\ {\text { volume must be positive.) }} \\ {\text { (c) Draw a graph of the function } V \text { and use it to estimate }} \\ {\text { the maximum volume for such a box. }}\end{array}
$$

Peter W.

Numerade Educator

Problem 80

Volume of a Box A cardboard box has a square base, with each edge of the base having length $x$ inches, as shown in the figure. The total length of all 12 edges of the box is
144 in.
(a) Show that the volume of the box is given by the function $V(x)=2 x^{2}(18-x)$.
(b) What is the domain of $V ?$ (Use the fact that length and volume must be positive.)
(c) Draw a graph of the function $V$ and use it to estimate the maximum volume for such a box.

Zach S.

Numerade Educator

Problem 81

Graphs of Large Powers Graph the functions $y=x^{2}$ , $y=x^{3}, y=x^{4},$ and $y=x^{5},$ for $-1 \leq x \leq 1,$ on the same coordinate axes. What do you think the graph of $y=x^{100}$ would look like on this same interval? What about $y=x^{101} ?$
Make a table of values to confirm your answers.

Peter W.

Numerade Educator

Problem 82

Maximum Number of Local Extrema What is the smallest possible degree that the polynomial whose graph is shown can have? Explain.

Peter W.

Numerade Educator

Problem 83

Possible Number of Local Extrema Is it possible for a third-degree polynomial to have exactly one local
extremum? Can a fourth-degree polynomial have exactly two local extrema? How many local extrema can polynomials of third, fourth, fifth, and sixth degree have? (Think about the end behavior of such polynomials.) Now give an example of a polynomial that has six local extrema.

Check back soon!

Problem 84

Impossible Situation? Is it possible for a polynomial to have two local maxima and no local minimum? Explain.

Olivia D.

Numerade Educator