Deck 3: Polynomial and Rational Functions

Question 1

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
$$f ( x ) = 3 - \frac { 1 } { 3 } x$$

A) Yes; degree 1
B) No; x has a fractional coefficient
C) Yes; degree 3
D) Yes; degree 0

Answer 1

A

Question 2

f(x) = (x + 2)5 + 4 A)

B)

C)

D)

Answer 2

C

Question 3

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
f(x) = 4x + 2x³

A) Yes; degree 3
B) Yes; degree 2
C) Yes; degree 1
D) Yes; degree 4

Answer 3

A

Question 4

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
$$f ( x ) = \frac { 2 - x ^ { 2 } } { 5 }$$

A) No; x is a negative term
B) Yes; degree 2
C) Yes; degree 1
D) No; it is a ratio

Question 5

f(x) = (x + 5)5 A)

B)

C)

D)

Question 6

A)

B)

C)

D)

Question 7

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
f(x) = 19x⁵ + 9x⁴ + 2

A) Yes; degree 9
B) No; the last term has no variable
C) Yes; degree 10
D) Yes; degree 5

Question 8

f(x) = x⁵ + 4 A)

B)

C)

D)

Question 9

$$f ( x ) = \frac { 1 } { 2 } ( x + 2 ) ^ { 4 } + 2$$ A)

B)

C)

D)

Question 10

A)

B)

C)

D)

Question 11

A)

B)

C)

D)

Question 12

f(x) = (x - 3)4 + 4 A)

B)

C)

D)

Question 13

$$f ( x ) = \frac { 1 } { 2 } ( x - 3 ) ^ { 5 } + 2$$

A)

B)

C)

D)

Question 14

A)

B)

C)

D)

Question 15

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
$$f ( x ) = 1 + \frac { 7 } { x }$$

A) Yes; degree 7
B) Yes; degree 0
C) Yes; degree 1
D) No; x is raised to a negative power

Question 16

A)

B)

C)

D)

Question 17

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
f(x) = x(x - 9)

A) Yes; degree 1
B) Yes; degree 2
C) Yes; degree 0
D) No; it is a product

Question 18

f(x) = 3 - (x - 4)4 A)

B)

C)

D)

Question 19

f(x) = -2(x - 5)4 + 3 A)

B)

C)

D)

Question 20

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
f(x) = 15

A) No; it contains no variables
B) No; it is a constant
C) Yes; degree 1
D) Yes; degree 0

Question 21

Determine the maximum number of turning points of f.
f(x) = -x² (x + 4)3(x² - 1)

A) 6
B) 2
C) 5
D) 7

Question 22

Find the x- and y-intercepts of f.
$$f ( x ) = 6 x - x ^ { 3 }$$

A) x-intercepts: $0 , \sqrt { 6 } , - \sqrt { 6 } ; y$ -intercept: 0
B) x-intercepts: $0 , - 6 ; y$ -intercept: 0
C) x-intercepts: $0 , - 6 ; y$ -intercept: 6
D) $x$ -intercepts: $0 , \sqrt { 6 } , - \sqrt { 6 } ; y$ -intercept: 6

Question 23

$$f ( x ) = 5 \left( x ^ { 2 } + 6 \right) \left( x ^ { 2 } + 4 \right) ^ { 2 }$$

A) $\sqrt { 6 }$ , multiplicity 1 , crosses $x$ -axis; $- \sqrt { 6 }$ , multiplicity 1 , crosses $x$ -axis; 2 , multiplicity 2 , touches $x$ -axis; $- 2$ , multiplicity 2 , touches $x$ -axis
B) -6, multiplicity 1 , touches $x$ -axis; $- 4$ , multiplicity 2, crosses $x$ -axis
C) -6, multiplicity 1 , crosses $x$ -axis; $- 4$ , multiplicity 2 , touches $x$ -axis
D) No real zeros

Question 24

Determine the maximum number of turning points of f.
$$f ( x ) = \left( x + \frac { 1 } { 9 } \right) ^ { 4 } ( x - 4 ) ^ { 3 }$$

A) above the $x$ -axis: $( 4 , \infty )$
below the $x$ -axis: $\left( - \infty , - \frac { 1 } { 9 } \right) , \left( - \frac { 1 } { 9 } , 4 \right)$
B) above the $x$ -axis: $\left( - \infty , - \frac { 1 } { 9 } \right) , \left( - \frac { 1 } { 9 } , 4 \right)$
below the $x$ -axis: $( 4 , \infty )$
C) above the $x$ -axis: $\left( - \frac { 1 } { 9 } , 4 \right)$
below the $x$ -axis: $\left( - \infty , - \frac { 1 } { 9 } \right) , ( 4 , \infty )$
D) above the x-axis: $\left( - \infty , - \frac { 1 } { 9 } \right) , ( 4 , \infty )$
below the $x$ -axis: $\left( - \frac { 1 } { 9 } , 4 \right)$

Question 25

Find the x- and y-intercepts of f.
$$f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } - 1 \right)$$

A) x-intercepts: -6, 0, 1; y-intercept: -6
B) x-intercepts: -1, 0, 1, 6; y-intercept: 0
C) x-intercepts: -6, -1, 0, 1; y-intercept: 0
D) x-intercepts: -6, -1, 0, 1; y-intercept: -6

Question 26

Find the x- and y-intercepts of f.
f(x) = (x - 3)(x - 1)

A) x-intercepts: 3, 1; y-intercept: -4
B) x-intercepts: -3, -1; y-intercept: -4
C) x-intercepts: 3, 1; y-intercept: 3
D) x-intercepts: -3, -1; y-intercept: 3

Question 27

Determine the maximum number of turning points of f.
f(x) = 8x - x³

A) 1
B) 3
C) 2
D) 4

Question 28

$$f ( x ) = \frac { 1 } { 3 } x ^ { 2 } \left( x ^ { 2 } - 5 \right) ( x - 3 )$$

A) 0 , multiplicity 2 , touches $x$ -axis;
3 , multiplicity 1 , crosses $x$ -axis;
$\sqrt { 5 }$ , multiplicity 1 , crosses $x$ -axis
$- \sqrt { 5 }$ , multiplicity 1 , crosses $x$ -axis
B) 0 , multiplicity 2 , crosses $x$ -axis;
3 , multiplicity 1 , touches $x$ -axis;
$\sqrt { 5 }$ , multiplicity 1 , touches $x$ -axis;
$- \sqrt { 5 }$ , multiplicity 1 , touches $x$ -axis
C) 0 , multiplicity 2, crosses x-axis;
3 , multiplicity 1 , touches $x$ -axis
D) 0 , multiplicity 2 , touches $x$ -axis;
3 , multiplicity 1 , crosses $x$ -axis

Question 29

$$f ( x ) = 4 - ( x + 4 ) ^ { 5 }$$

A)

B)

C)

D)

Question 30

$$f ( x ) = - 2 ( x - 5 ) ^ { 5 } + 3$$

A)

B)

C)

D)

Question 31

Find the x- and y-intercepts of f.
$$f ( x ) = 2 x ^ { 4 } ( x - 5 ) ^ { 5 }$$

A) x-intercepts: 0, -5; y-intercept: 2
B) x-intercepts: 0, 5; y-intercept: 2
C) x-intercepts: 0, -5; y-intercept: 0
D) x-intercepts: 0, 5; y-intercept: 0

Question 32

Determine the maximum number of turning points of f.
f(x) = (x - 3)²(x + 5)²

A) 3
B) 1
C) 4
D) 2

Question 33

Find the x- and y-intercepts of f.
$$f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$$ $$( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$$

A) x-intercepts: -6, -1, 0; y-intercept: 6
B) x-intercepts: -6, -1, 0; y-intercept: -6
C) x-intercepts: -6, -1, 0, 1; y-intercept: 0
D) x-intercepts: -6, 0; y-intercept: 0

Question 34

Determine the maximum number of turning points of f.
$$f ( x ) = x ^ { 2 } ( x + 2 )$$
Answer: (a) For large values of $| x |$ , the graph of $f ( x )$ will resemble the graph of $y = x ^ { 3 }$ .
(b) $y$ -intercept: $( 0,0 )$ , $x$ -intercepts: $( 0,0 )$ and $( - 2,0 )$
(c) The graph of $f$ crosses the $x$ -axis at $( - 2,0 )$ and touches the $x$ -axis at $( 0,0 )$ .
(e) Local minimum at $( 0,0 )$ , Local maximum at $( - 1.33,1.19 )$
(f)

(g) Domain of f: all real numbers; range of f: all real numbers
(h) f is increasing on (-, -1.33) and (0, ); f is decreasing on (-1.33, 0)

Question 35

$$f ( x ) = \frac { 1 } { 5 } x ^ { 4 } \left( x ^ { 2 } - 5 \right)$$

A) 0 , multiplicity 4 , touches $x$ -axis;
$\sqrt { 5 }$ , multiplicity 1 , crosses $x$ -axis;
$- \sqrt { 5 }$ , multiplicity 1 , crosses $x$ -axis
B) 0 , multiplicity 4 , crosses $x$ -axis
C) 0 , multiplicity 4 , crosses $x$ -axis;
$\sqrt { 5 }$ , multiplicity 1 , touches $x$ -axis;
D) 0 , multiplicity 4 , touches $x$ -axis
$- \sqrt { 5 }$ , multiplicity 1 , touches $x$ -axis

Question 36

Determine the maximum number of turning points of f.
$$f ( x ) = ( x - 4 ) ^ { 3 }$$

A) above the $x$ -axis: $( 4 , \infty )$
below the $x$ -axis: $( - \infty , 4 )$
B) above the $x$ -axis: $( - \infty , 4 ) , ( 4 , \infty )$
below the $x$ -axis: no intervals
C) above the x-axis: no intervals
below the $x$ -axis: $( - \infty , 4 ) , ( 4 , \infty )$
D) above the $x$ -axis: $( - \infty , 4 )$
below the $x$ -axis: $( 4 , \infty )$

Question 37

Find the x- and y-intercepts of f.
$$f ( x ) = ( x + 5 ) ( x - 3 ) ( x + 3 )$$

A) x-intercepts: -3, 3, 5; y-intercept: -45
B) x-intercepts: -5, -3, 3; y-intercept: -45
C) x-intercepts: -5, -3, 3; y-intercept: 45
D) x-intercepts: -3, 3, 5; y-intercept: 45

Question 38

Find the x- and y-intercepts of f.
$$f ( x ) = ( x + 1 ) ( x - 2 ) ( x - 1 ) ^ { 2 }$$

A) x-intercepts: -1, 1, -2; y-intercept: 2
B) x-intercepts: -1, 1, 2; y-intercept: 2
C) x-intercepts: -1, 1, -2; y-intercept: -2
D) x-intercepts: -1, 1, 2; y-intercept: -2

Question 39

Find the x- and y-intercepts of f.
$$f ( x ) = ( x + 6 ) ^ { 2 }$$

A) x-intercept: -6; y-intercept: 36
B) x-intercept: -6; y-intercept: 0
C) x-intercept: 6; y-intercept: 36
D) x-intercept: 6; y-intercept: 0

Question 40

Determine the maximum number of turning points of f.
$$f ( x ) = ( x - 3 ) ^ { 2 } ( x + 5 ) ^ { 2 }$$

A) above the $x$ -axis: $( - \infty , - 5 ) , ( 3 , \infty )$
below the x-axis: $( - 5,3 )$
B) above the x-axis: $( - 5,3 )$
below the $x$ -axis: $( - \infty , - 5 ) , ( 3 , \infty )$
C) above the $x$ -axis: $( - \infty , - 5 ) , ( - 5,3 ) , ( 3 , \infty )$
below the $x$ -axis: no intervals
D) above the $x$ -axis: no intervals
below the $x$ -axis: $( - \infty , - 5 ) , ( - 5,3 ) , ( 3 , \infty )$

Question 41

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the
zeros to factor f over the real numbers.
$$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12$$

A) $- 3 ; f ( x ) = ( x + 3 ) \left( x ^ { 2 } - x - 4 \right)$
B) $- 3 , - 2,2 ; f ( x ) = ( x + 3 ) ( x + 2 ) ( x - 2 )$
C) $- 2 ; f ( x ) = ( x + 2 ) \left( x ^ { 2 } + x - 6 \right)$
D) $- 2,2,3 ; f ( x ) = ( x + 2 ) ( x - 2 ) ( x - 3 )$

Question 42

List the potential rational zeros of the polynomial function. Do not find the zeros.
$$f ( x ) = x ^ { 5 } - 4 x ^ { 2 } + 3 x + 14$$

A) $\pm 1 , \pm 7 , \pm 2 , \pm 14$
B) $\pm 1 , \pm \frac { 1 } { 7 } , \pm \frac { 1 } { 2 } , \pm \frac { 1 } { 14 }$
C) $\pm 1 , \pm 7 , \pm 2$
D) $\pm 1 , \pm \frac { 1 } { 7 } , \pm \frac { 1 } { 2 } , \pm \frac { 1 } { 14 } , \pm 7 , \pm 2 , \pm 14$

Question 43

Find the x- and y-intercepts of f.
$$f ( x ) = ( x + 13 ) ^ { 2 }$$

A) x-intercept: 13; y-intercept: 169
B) x-intercept: -13; y-intercept: 0
C) x-intercept: -13; y-intercept: 169
D) x-intercept: 13; y-intercept: 0

Question 44

List the potential rational zeros of the polynomial function. Do not find the zeros.
$$f ( x ) = 6 x ^ { 4 } + 4 x ^ { 3 } - 2 x ^ { 2 } + 2$$

A) $\pm \frac { 1 } { 2 } , \pm \frac { 3 } { 2 } , \pm 1 , \pm 2 , \pm 3 , \pm 6$
B) $\pm \frac { 1 } { 6 } , \pm \frac { 1 } { 3 } , \pm \frac { 1 } { 2 } , \pm \frac { 2 } { 3 } , \pm 1 , \pm 2$
C) $\pm \frac { 1 } { 6 } , \pm \frac { 1 } { 3 } , \pm \frac { 1 } { 2 } , \pm \frac { 2 } { 3 } , \pm 1 , \pm 2 , \pm 3$
D) $\pm \frac { 1 } { 6 } , \pm \frac { 1 } { 3 } , \pm \frac { 1 } { 2 } , \pm 1 , \pm 2$

Question 45

List the potential rational zeros of the polynomial function. Do not find the zeros.
$$f ( x ) = x ^ { 5 } - 5 x ^ { 2 } + 2 x + 7$$

A) $\pm 1 , \pm \frac { 1 } { 7 }$
B) $\pm 7 , \pm \frac { 1 } { 7 }$
C) $\pm \frac { 1 } { 5 } , \pm \frac { 7 } { 5 } , \pm 7$
D) $\pm 1 , \pm 7$

Question 46

Solve the problem.
The profits (in millions) for a company for 8 years was as follows: $$\begin{array} { c | c } \text { Year, } x & \text { Profits } \\\hline 1993,1 & 1.1 \\1994,2 & 1.7 \\1995,3 & 2.0 \\1996,4 & 1.4 \\1997,5 & 1.3 \\1998,6 & 1.5 \\1999,7 & 1.8 \\2000,8 & 2.1\end{array}$$ Find the cubic function of best fit to the data.

Question 47

Solve the problem.
For the polynomial funct $$f ( x ) = 2 x ^ { 4 } - 7 x ^ { 3 } + 11 x - 4$$ a) Find the x- and y-intercepts of the graph of f. Round to two decimal places, if necessary.
b) Determine whether the graph crosses or touches the x-axis at each x-intercept.
c) End behavior: find the power function that the graph of f resembles for large values of |x|.
d) Use a graphing utility to graph the function.Approximate the local maxima rounded to two decimal places,
if necessary. Approximate the local minima rounded to two decimal places, if necessary.
e) Determine the number of turning points on the graph.
f) Put all the information together, and connect the points with a smooth, continuous curve to obtain the
graph of f.

Question 48

Find the x- and y-intercepts of f.
f(x) = (x + 6)(x - 2)(x + 2)

A) x-intercepts: -6, -2, 2; y-intercept: -24
B) x-intercepts: -2, 2, 6; y-intercept: 24
C) x-intercepts: -2, 2, 6; y-intercept: -24
D) x-intercepts: -6, -2, 2; y-intercept: 24

Question 49

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the
zeros to factor f over the real numbers.
$$f ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + 9 x - 6$$

A) $6 ; f ( x ) = ( x - 6 ) \left( 3 x ^ { 2 } + 1 \right)$
B) $3 , \frac { 2 } { 3 } , 1 ; f ( x ) = ( 3 x - 2 ) ( x - 1 ) ( x - 3 )$
C) $\frac { 2 } { 3 } ; f ( x ) = ( 3 x - 2 ) \left( x ^ { 2 } + 3 \right)$
D) $- 3 , - 1 , \frac { 2 } { 3 } ; f ( x ) = ( 3 x - 2 ) ( x + 1 ) ( x + 3 )$

Question 50

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the
zeros to factor f over the real numbers.
$$f ( x ) = x ^ { 4 } - 15 x ^ { 2 } - 16$$

A) $- 4,4 ; f ( x ) = ( x - 4 ) ( x + 4 ) \left( x ^ { 2 } + 1 \right)$
B) $4 ; f ( x ) = ( x - 4 ) ^ { 2 } \left( x ^ { 2 } + 1 \right)$
C) $- 1,1 ; f ( x ) = ( x - 1 ) ( x + 1 ) \left( x ^ { 2 } + 16 \right)$
D) $- 4 , - 1,4,1 ; f ( x ) = ( x - 4 ) ( x + 4 ) ( x - 1 ) ( x + 1 )$

Question 51

Find the x- and y-intercepts of f.
f(x) = 4x² (x - 4)5

A) x-intercepts: 0, -4; y-intercept: 0
B) x-intercepts: 0, 4; y-intercept: 4
C) x-intercepts: 0, -4; y-intercept: 4
D) x-intercepts: 0, 4; y-intercept: 0

Question 52

Solve the problem.
The amount of water (in gallons) in a leaky bathtub is given in the table below. Using a graphing utility, fit the data to a third degree polynomial (or a cubic). Then approximate the time at which there is maximum amount
Of water in the tub, and estimate the time when the water runs out of the tub. Express all your answers rounded
To two decimal places. $$\begin{array} { c | c c c c c c c c } \mathrm { t } \text { (in minutes) } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \mathrm { V } \text { (in gallons) } & 20 & 26 & 45 & 63 & 86 & 94 & 90 & 67\end{array}$$

A) maximum amount of water after 8.23 minutes; water runs out after 19.73 minutes
B) maximum amount of water after 5.31 minutes; water runs out after 8.23 minutes
C) maximum amount of water after 5.37 minutes; water runs out after 11.06 minutes
D) maximum amount of water after 5.31 minutes; water never runs out

Question 53

List the potential rational zeros of the polynomial function. Do not find the zeros.
$$f ( x ) = - 4 x ^ { 4 } + 2 x ^ { 2 } - 3 x + 6$$

A) $\pm \frac { 1 } { 6 } , \pm \frac { 1 } { 2 } , \pm \frac { 1 } { 3 } , \pm \frac { 2 } { 3 } , \pm \frac { 4 } { 3 } , \pm 1 , \pm 2 , \pm 4$
B) $\pm \frac { 1 } { 4 } , \pm \frac { 1 } { 2 } , \pm \frac { 3 } { 4 } , \pm \frac { 3 } { 2 } , \pm 1 , \pm 2 , \pm 3 , \pm 4 , \pm 6$
C) $\pm \frac { 1 } { 4 } , \pm \frac { 1 } { 2 } , \pm \frac { 3 } { 4 } , \pm \frac { 3 } { 2 } , \pm 1 , \pm 2 , \pm 3 , \pm 6$
D) $\pm \frac { 1 } { 4 } , \pm \frac { 1 } { 2 } , \pm \frac { 2 } { 3 } , \pm \frac { 3 } { 4 } , \pm \frac { 3 } { 2 } , \pm 1 , \pm 2 , \pm 3 , \pm 6$

Question 54

List the potential rational zeros of the polynomial function. Do not find the zeros.
$$f ( x ) = 2 x ^ { 5 } - 5 x ^ { 2 } + 3 x - 1$$

A) $\pm 1 , \pm 2 , \pm \frac { 1 } { 2 }$
B) $\pm 1$ , $\pm 2$
C) $\pm 2 , \pm \frac { 1 } { 2 }$
D) $\pm 1 , \pm \frac { 1 } { 2 }$

Question 55

Solve the problem.
Which of the following polynomial functions might have the graph shown in the illustration below?

A) $f ( x ) = x ( x - 2 ) ^ { 2 } ( x - 1 )$
B) $f ( x ) = x ^ { 2 } ( x - 2 ) ^ { 2 } ( x - 1 ) ^ { 2 }$
C) $f ( x ) = x ^ { 2 } ( x - 2 ) ( x - 1 )$
D) $f ( x ) = x ( x - 2 ) ( x - 1 ) ^ { 2 }$

Question 56

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the
zeros to factor f over the real numbers.
$$f ( x ) = 5 x ^ { 4 } - 7 x ^ { 3 } + 22 x ^ { 2 } - 28 x + 8$$

A) $- 4 , - 1,1 , - \frac { 2 } { 5 } ; f ( x ) = ( x - 1 ) ( 5 x + 2 ) ( x + 1 ) ( x + 4 )$
B) $- 4 , - 1,1 , \frac { 2 } { 5 } ; f ( x ) = ( x - 1 ) ( 5 x - 2 ) ( x + 1 ) ( x + 4 )$
C) $4 , \frac { 2 } { 5 } ; f ( x ) = ( x - 4 ) ( 5 x - 2 ) \left( x ^ { 2 } + 1 \right)$
D) $1 , \frac { 2 } { 5 } ; f ( x ) = ( x - 1 ) ( 5 x - 2 ) \left( x ^ { 2 } + 4 \right)$

Question 57

List the potential rational zeros of the polynomial function. Do not find the zeros.
$$f ( x ) = 11 x ^ { 3 } - x ^ { 2 } + 3$$

A) $\pm \frac { 1 } { 11 } , \pm \frac { 3 } { 11 } , \pm 1 , \pm 3 , \pm 11$
B) $\pm \frac { 1 } { 11 } , \pm \frac { 3 } { 11 } , \pm 1 , \pm 3$
C) $\pm \frac { 1 } { 11 } , \pm \frac { 1 } { 3 } , \pm 1 , \pm 3 , \pm 11$
D) $\pm \frac { 1 } { 3 } , \pm \frac { 11 } { 3 } , \pm 1 , \pm 11$

Question 58

List the potential rational zeros of the polynomial function. Do not find the zeros.
$$f ( x ) = - 2 x ^ { 3 } + 2 x ^ { 2 } - 3 x + 8$$

A) $\pm \frac { 1 } { 8 } , \pm \frac { 1 } { 4 } , \pm \frac { 1 } { 2 } , \pm 1 , \pm 2 , \pm 4 , \pm 8$
B) $\pm \frac { 1 } { 2 } , \pm 1 , \pm 2 , \pm 4$
C) $\pm \frac { 1 } { 4 } , \pm \frac { 1 } { 2 } , \pm 1 , \pm 2 , \pm 4 , \pm 8$
D) $\pm \frac { 1 } { 2 } , \pm 1 , \pm 2 , \pm 4 , \pm 8$

Question 59

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the
zeros to factor f over the real numbers.
$$f ( x ) = 2 x ^ { 4 } - 12 x ^ { 3 } + 19 x ^ { 2 } - 6 x + 9$$

A) $- 3,3 ; f ( x ) = ( x - 3 ) ( x + 3 ) \left( 2 x ^ { 2 } + 1 \right)$
B) no real roots; $f ( x ) = \left( x ^ { 2 } + 9 \right) \left( 2 x ^ { 2 } + 1 \right)$
C) 3 , multiplicity $2 ; f ( x ) = ( x - 3 ) ^ { 2 } \left( 2 x ^ { 2 } + 1 \right)$
D) $- 3$ , multiplicity $2 ; f ( x ) = ( x + 3 ) ^ { 2 } \left( 2 x ^ { 2 } + 1 \right)$

Question 60

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the
zeros to factor f over the real numbers.
$$f ( x ) = 2 x ^ { 3 } + 13 x ^ { 2 } + 10 x + 25$$

A) $- 1 , \frac { 5 } { 2 } , 5 ; f ( x ) = ( 2 x - 5 ) ( x - 5 ) ( x + 1 )$
B) $- 5 , \frac { 5 } { 2 } , 1 ; f ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x + 5 )$
C) $1 , \frac { 2 } { 5 } , - 5 ; f ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x + 5 )$
D) $- 1 , \frac { 2 } { 5 } , - 5 ; f ( x ) = ( 2 x - 5 ) ( x - 5 ) ( x + 1 )$

Question 61

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places.
x³ - 8x - 3 = 0; -1 ≤ r ≤ 0

Question 62

Find the x- and y-intercepts of f.
f(x) = 6x - x³

A) x-intercepts: 0, -6; y-intercept: 0
B) x-intercepts: 0, 6, -6; y-intercept: 0
C) x-intercepts: 0, 6, - 6; y-intercept: 6
D) x-intercepts: 0, -6; y-intercept: 6

Question 63

Find the x- and y-intercepts of f.
f(x) = (x - 2)²(x² - 9)

A) x-intercepts: -3, 2, 3; y-intercept: 36
B) x-intercepts: -2, -9; y-intercept: 18
C) x-intercepts: 2, 9; y-intercept: 18
D) x-intercepts: -3, 2, 3; y-intercept: -36

Question 64

Solve the equation in the real number system.
$2 x ^ { 3 } - x ^ { 2 } + 2 x - 1 = 0$

A) $\left\{ - 2 , - \frac { 1 } { 2 } , - 1 \right\}$
B) $\left\{ \frac { 1 } { 2 } \right\}$
C) $\left\{ - 2 , \frac { 1 } { 2 } , - 1 \right\}$
D) $\left\{ \frac { 1 } { 2 } , - 1 \right\}$

Question 65

Solve the equation in the real number system.
$$x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24 = 0$$

A) {-4, -2}
B) {2, 4}
C) {-3, 2, 4}
D) {-4, -2, 3}

Question 66

Find the x- and y-intercepts of f.
f(x) = -x² (x + 7)(x² + 1)

A) x-intercepts: -7, 0; y-intercept: 0
B) x-intercepts: -7, -1, 0; y-intercept: 7
C) x-intercepts: -7, -1, 0, 1; y-intercept: 0
D) x-intercepts: -7, -1, 0; y-intercept: -7

Question 67

Solve the equation in the real number system.
$$x ^ { 3 } + 8 x ^ { 2 } - 18 x + 20 = 0$$

A) {1}
B) {10}
C) {-10, 10}
D) {-10}

Question 68

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places.
x³ - 8x - 3 = 0; -3 ≤ r ≤-2

Question 69

Solve the equation in the real number system.
$2 x ^ { 4 } - 2 x ^ { 3 } + x ^ { 2 } - 5 x - 10 = 0$

A) $\left\{ - \frac { \sqrt { 10 } } { 2 } , \frac { \sqrt { 10 } } { 2 } \right\}$
B) $\{ 1 , - 2 \}$
C) $\{ - 1,2 \}$
D) $\left\{ - \frac { 5 } { 2 } , \frac { 5 } { 2 } \right\}$

Question 70

Solve the equation in the real number system.
$3 x ^ { 4 } - 29 x ^ { 3 } + 111 x ^ { 2 } - 179 x + 78 = 0$

A) $\left\{ - 3 , - \frac { 2 } { 3 } \right\}$
B) $\left\{ - 3 , \frac { 2 } { 3 } \right\}$
C) $\left\{ 3 , - \frac { 2 } { 3 } \right\}$
D) $\left\{ 3 , \frac { 2 } { 3 } \right\}$

Question 71

Solve the equation in the real number system.
$$x ^ { 4 } - 12 x ^ { 2 } - 64 = 0$$

A) {-8, 8}
B) {-4, 4}
C) {-2, 2}
D) {-4, -2, 2, 4}

Question 72

Find the x- and y-intercepts of f.
f(x) = x² (x - 1)(x - 3)

A) x-intercepts: 0, -1, -3; y-intercept: 3
B) x-intercepts: 0, 1, 3; y-intercept: 3
C) x-intercepts: 0, 1, 3; y-intercept: 0
D) x-intercepts: 0, -1, -3; y-intercept: 0

Question 73

Solve the equation in the real number system.
$$x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - x - 10 = 0$$

A) {-1, 2}
B) {-1, -2}
C) {1, 2}
D) {-2, 1}

Question 74

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places.
x⁴ - x³ - 7x² + 5x + 10 = 0; -3 ≤ r ≤-2

Question 75

Find the x- and y-intercepts of f.
f(x) = (x + 1)(x - 4)(x - 1)²

A) x-intercepts: -1, 1, -4; y-intercept: -4
B) x-intercepts: -1, 1, 4; y-intercept: 4
C) x-intercepts: -1, 1, 4; y-intercept: -4
D) x-intercepts: -1, 1, -4; y-intercept: 4

Question 76

Find the x- and y-intercepts of f.
f(x) = -x² (x + 2)(x² - 1)

A) x-intercepts: -2, 0, 1; y-intercept: -2
B) x-intercepts: -2, -1, 0, 1; y-intercept: -2
C) x-intercepts: -1, 0, 1, 2; y-intercept: 0
D) x-intercepts: -2, -1, 0, 1; y-intercept: 0

Question 77

Find the x- and y-intercepts of f.
f(x) = (x - 2)(x - 5)

A) x-intercepts: -2, -5; y-intercept: 10
B) x-intercepts: 2, 5; y-intercept: -7
C) x-intercepts: 2, 5; y-intercept: 10
D) x-intercepts: -2, -5; y-intercept: -7

Question 78

Solve the equation in the real number system.
$$2 x ^ { 3 } - 13 x ^ { 2 } + 22 x - 8 = 0$$

A) $\left\{ \frac { 1 } { 2 } , 2,4 \right\}$
B) $\{ - 2,1 , - 2 \rangle$
C) $\{ 2,1,2 \}$
D) $\left\{ - \frac { 1 } { 2 } , 2 , - 4 \right\}$

Question 79

Solve the equation in the real number system.
$$x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 8 x - 17 = 0$$

A) {-1, 4}
B) {-1, 1}
C) {-4, 4}
D) {-4, 1}

Question 80

Solve the equation in the real number system.
$$2 x ^ { 3 } - x ^ { 2 } - 6 x + 3 = 0$$

A) $\{ 2 , \sqrt { 3 } , - \sqrt { 3 } \}$
B) $\left\{ - \frac { 1 } { 2 } , \sqrt { 3 } , - \sqrt { 3 } \right\}$
C) $\left\{ \frac { 1 } { 2 } , \sqrt { 3 } , - \sqrt { 3 } \right\}$
D) $\{ - 2 , \sqrt { 3 } , - \sqrt { 3 } \}$