Deck 4: Exponential and Logarithmic Functions

Question 1

Find all rational, irrational, and complex zeros (and state their multiplicities). $P ( x ) = x ^ { 3 } - 8$

Answer 1

$2 , - 1 - \sqrt { 3 } i , - 1 + \sqrt { 3 } i$

Question 2

Use a graphing device to graph the polynomial $y = 2 x ^ { 5 } + 3 x ^ { 2 } - 2 x + 1$
and describe its end behavior.

Answer 2

$y \rightarrow \infty$ as $x \rightarrow \infty$ ;
$y \rightarrow - \infty$ as $x \rightarrow - \infty$

Question 3

Find the indicated value of the polynomial using the Remainder Theorem. $p ( x ) = 2 x ^ { 3 } - 2 x ^ { 2 } + 11 x - 100$
; find $p ( 3 )$

Answer 3

$\begin{array}{rrrrr}3 & 2 & -2 & 11 & -100 \\\hline && 6 & 12 & 69 \\\hline& 2 & 4 & 23 & -31\end{array}$

; Therefore, $p ( 3 ) = - 31$

Question 4

Find the complete factorization of

Question 5

Evaluate the expression and write the result in the form a + bi.

Question 6

If a ball is thrown directly upward with a velocity of 64 ft/s, its height (in feet) after t seconds is givenby $y = 64 t - 16 t ^ { 2 }$ . What is the maximum height attained by the ball?

Question 7

Find the quotient and remainder using long division. $\frac { x ^ { 6 } + x ^ { 2 } + 4 x ^ { 4 } + 4 } { x ^ { 2 } + 4 }$

Question 8

Find all real and complex zeros of $P ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 6 x - 8$

Question 9

Evaluate the expression and write in the form $a + b i$ . $i ^ { 37 }$

Question 10

Use the Factor Theorem to show that
is a factor of the polynomial.

Question 11

Evaluate the expression $\sqrt { - 10 } \cdot \sqrt { - 40 }$ and write in the form $a + b i$

Question 12

Find the remainder if $p ( x ) = 6 x ^ { 342 } + 18 x ^ { 63 } - 5 x ^ { 2 } - 20$ is divided by $( x - 1 )$

Question 13

Find a polynomial of degree $4$ , with integer coefficients that has zeros of $3 i$
and $2$ , with $2$ a double zero. (There are many possibilities.)

Question 14

Find a polynomial of degree 3 that has zeros 1, 2, and -3 and in which the coefficient of $x$ is -2.

Question 15

List all possible rational roots. $x ^ { 5 } - 4 x ^ { 4 } - 13 x ^ { 3 } + 70 x ^ { 2 } - 90 x + 36$

Question 16

Find all rational, irrational, and complex zeros (and state their multiplicities). $P ( x ) = 2 x ^ { 3 } - 12 x ^ { 2 } + 24 x - 16$

Question 17

Find the quotient and remainder. $\frac { x ^ { 3 } + 3 x - 2 } { x - 1 }$

Question 18

Evaluate the expression and write the result in the form a + bi. $\frac { 36 } { 6 - 6 i }$

Question 19

Sketch the graph of the function $y = ( x - 3 ) ^ { 2 }$ by transforming the graph of an appropriate function of the form $y = x ^ { n }$ . Indicate all $x$ - and $y -$ intercepts on your graph.

Question 20

A quadratic function is given.
(a) Express the quadratic function in standard form
b) Find its vertex and its x- and y-intercept(s). $f ( x ) = 2 x ^ { 2 } + 8 x + 9$

Question 21

Find a polynomial of degree 3 with constant coefficient $- 18$
and zeros $- 1$ , $- 2$ , and $3$
.

Question 22

Find the indicated value of the polynomial using the Remainder Theorem. $p ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 76$
; find $p ( - 4 )$

Question 23

Graph the rational function. Show clearly all x- and y-intercepts and asymptotes.

Question 24

Find the quotient and remainder using long division.

Question 25

Find all horizontal and vertical asymptotes of the function $y( x ) = \frac { 5 } { x - 5 }$

Question 26

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function,
where the revenue
is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

Question 27

Evaluate the expression and write the result in the form a + bi.

Question 28

Determine the number of possible positive and negative real zeros using Descartes' Rule of Signs.

Question 29

Graph the rational function. Show clearly all x- and y-intercepts and asymptotes. $y = \frac { 2 x + 4 } { x - 2 }$

Question 30

Sketch the graph of
to find all real zeros.

Question 31

Evaluate the expression and write the result in the form a + bi.

Question 32

A quadratic function is given. (a) Express the quadratic function in standard form
b) Find its vertex and its x- and y-intercept(s).

Question 33

Sketch the graph of the function
by transforming the graph of an appropriate function of the form
. Indicate all
- and
intercepts on your graph.

Question 34

What is the remainder when the polynomial
is divided by
?

Question 35

Use a graphing calculator to find all real solutions of the equation. $- 3 x = - \frac { 2 } { 3 } - 3 x ^ { 2 }$

Question 36

Find the quotient and remainder.

Question 37

Use a graphing device to graph the polynomial $P ( x ) = 3 x ^ { 4 } - 4 x ^ { 3 } - 10 x$
and describe its end behavior.

Question 38

Use the Factor Theorem to show that
is a factor of the polynomial.

Question 39

Find all horizontal and vertical asymptotes of the function $y ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 11 x + 30 }$

Question 40

If a ball is thrown directly upward with a velocity of 48 ft/s, its height (in feet) after t seconds is given
By
) What is the maximum height attained by the ball?

A)
B)
C)
D)
E) none of these

Question 41

Find a polynomial of degree 3 that has zeros 1, -2, and 3 and in which the coefficient of $x ^ { 2 }$ is -5.

Question 42

Express the quadratic function in standard form.

A)
B)
C)
D)
E) none of these

Question 43

Use a graphing device to graph the polynomial $p ( x ) = 3 x ^ { 4 } - 4 x ^ { 3 } - 10 x$ to describe its end behavior

A) $p ( x ) \rightarrow 0$ As $x \rightarrow \infty$
; $p ( x ) \rightarrow \infty$ As $x \rightarrow - \infty$

B) $p ( x ) \rightarrow \infty$ As $x \rightarrow \infty$
; $p ( x ) \rightarrow 0$ As $x \rightarrow - \infty$

C) $p ( x ) \rightarrow - \infty$ As $x \rightarrow \infty$
; $p ( x ) \rightarrow - \infty$ As $x \rightarrow - \infty 3$

D) $p ( x ) \rightarrow - \infty$ As $x \rightarrow \infty$
; $p ( x ) \rightarrow \infty$ As $x \rightarrow - \infty$

E) $p ( x ) \rightarrow \infty$ As $x \rightarrow \infty$
; $p ( x ) \rightarrow \infty$ As $x \rightarrow - \infty$

Question 44

Find all rational, irrational, and complex zeros (and state their multiplicities). $P ( x ) = x ^ { 4 } - 16$

Question 45

Find the indicated value of the polynomial using the Remainder Theorem. ; find

A)
B)
C)
D)
E)

Question 46

Find all horizontal and vertical asymptotes of the function
.

Question 47

Graph the rational function. Show clearly all x- and y-intercepts and asymptotes.

Question 48

Find the complete factorization of

Question 49

Use a graphing calculator to find all real solutions of the equation. $x ^ { 5 } = x + 3$

Question 50

Find the quotient and remainder using long division.

A) Quotient:
; Remainder:
B) Quotient:
; Remainder:
C) Quotient:
; Remainder:
D) Quotient:
; Remainder:
E) none of these

Question 51

What is the remainder when the polynomial is divided by
?

A)
B)
C)
D)
E)

Question 52

Divide.

A)
B)
C)
D)
E) none

Question 53

Find all horizontal and vertical asymptotes of the function
.

Question 54

Which of the following is a factor of the polynomial?

A)
B)
C)
D)
E)

Question 55

Sketch the graph of the function by transforming the graph of an appropriate function of the form
) Indicate all
- and
Intercepts on your graph

A)

B)

C)

D)

Question 56

Determine the number of possible positive and negative real zeros using Descartes' Rule of Signs.

A) Positive real: exactly one; Negative real: exactly one
B) Positive real: 2 or 0; Negative real: exactly 2
C) Positive real: 4 or 0; Negative real: exactly one
D) Positive real: exactly 4; Negative real: exactly one
E) Positive real: 4, 2, or 0; Negative real: exactly one

Question 57

Use a graphing device to graph the polynomial $p ( x ) = - x ^ { 5 } - 4 x ^ { 3 } - 10 x$ to describe its end behavior

A) $p ( x ) \rightarrow 0$ As $x \rightarrow \infty$
; $p ( x ) \rightarrow \infty$ As $x \rightarrow - \infty$

B) $p ( x ) \rightarrow \infty$ As $x \rightarrow \infty$
; $p ( x ) \rightarrow 0$ As $x \rightarrow - \infty 3$

C) $p ( x ) \rightarrow - \infty$ As $x \rightarrow \infty$
; $p ( x ) \rightarrow \infty$ As $x \rightarrow - \infty$

D) $p ( x ) \rightarrow - 1$ As $x \rightarrow \infty$
; $p ( x ) \rightarrow \infty$ As $x \rightarrow - \infty$

E) $p ( x ) \rightarrow - \infty$ As $x \rightarrow \infty$
; $p ( x ) \rightarrow - \infty$ As $x \rightarrow - \infty$

Question 58

Find all horizontal and vertical asymptotes of the function .

A) b) c) d) e)

Question 59

Find the equation of the slant asymptote, if any.

A)
B)
C)
D)
E)

Question 60

Find a polynomial of degree $4$ , with integer coefficients that has zeros of $3 i$ And $2$ , with $2$ A double zero

A) $x ^ { 4 } + 4 x ^ { 3 } - 5 x ^ { 2 } - 36 x - 36$
B) $x ^ { 4 } + 13 x ^ { 2 } - 4 x ^ { 3 } - 36 x + 36$
C) $x ^ { 4 } + 13 x ^ { 2 } + 4 x ^ { 3 } + 36 x + 36$
D) $4 x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - 36 x - 36$
E) $x ^ { 4 } + 4 x ^ { 3 } + 5 x - 36$

Question 61

Find all rational, irrational, and complex zeros.

A)
B)
C)
D)
E)

Question 62

Sketch the graph of the function by transforming the graph of an appropriate function of the form

A)

B)

C)

D)

Question 63

Factor the polynomial completely.

A)
B)
C)
D)
E) none

Question 64

Sketch the graph of to find all real zeros.

A) zeros:
B) zeros:
C) zeros:
D) zeros:
E) zeros:

Question 65

Find all rational, irrational, and complex zeros.

A)
B)
C)
D)
E)

Question 66

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function, where the revenue
Is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

A) Maximum Revenue:
When
Units are manufactured
B) Maximum Revenue:
When
Units are manufactured
C) Maximum Revenue:
When
Units are manufactured
D) Maximum Revenue:
When
Units are manufactured
E) none of these

Question 67

Use a graphing calculator to find all real solutions of the equation.

A) b) c) d) e)

Question 68

Find a polynomial of degree 3 with constant coefficient $12$ and zeros $1 / 2$ , $- 2$ , and $3$

A) $4 x ^ { 3 } - 6 x ^ { 2 } - 22 x - 12$
B) $x ^ { 3 } - 6 x ^ { 2 } - 22 x - 12$
C) $4 x ^ { 3 } - 6 x ^ { 2 } - 22 x + 12$
D) $4 x ^ { 3 } - 6 x ^ { 2 } + 22 x + 12$
E) $4 x ^ { 3 } + 6 x ^ { 2 } - 22 x + 12$

Question 69

Divide. $\frac { x ^ { 3 } + 2 x ^ { 2 } - 10 } { x + 3 }$

A) $\frac { x ^ { 3 } + 2 x ^ { 2 } - 10 } { x + 3 } = x ^ { 2 } - x + 3 - \frac { 10 } { x + 3 }$

B) $\frac { x ^ { 3 } + 2 x ^ { 2 } - 10 } { x + 3 } = x ^ { 2 } - x + 2 - \frac { 19 } { x + 3 }$

C) $\frac { x ^ { 3 } + 2 x ^ { 2 } - 10 } { x + 3 } = \frac { x } { 3 } + \frac { 2 x } { 3 } - \frac { 10 } { x + 3 }$

D) $\frac { x ^ { 3 } + 2 x ^ { 2 } - 10 } { x + 3 } = x ^ { 2 } + x + 3 + \frac { 19 } { x + 3 }$

E) none

Question 70

Graph the rational function.

A)

B)
C)

D)

Question 71

Find all zeros.

A)
B)
C)
D)
E) none of these

Question 72

Find all vertical asymptotes of the function

A)
B)
C)
D)
E)

Question 73

Evaluate the expression and write in the form .

A)
B)
C)
D)
E)

Question 74

Evaluate the expression and write the result in the form a + bi.

A)
B)
C)
D)
E)

Question 75

Evaluate the expression and write in the form

A)
B)
C)
D)
E)

Question 76

Express the quadratic function in standard form.

A)
B)
C)
D)
E) none of these

Question 77

Evaluate the expression and write in the form

A)
B)
C)
D)
E)

Question 78

Evaluate the expression and write in the form .

A)
B)
C)
D)
E)

Question 79

Find the equation of the slant asymptote and the vertical asymptote, if any.

A)
B)
C)
D)
E)

Question 80

Find a polynomial of degree $4$ , with integer coefficients that has zeros of $3 i$ And $2$ , with $2$
A double zero

A) $x ^ { 4 } + 4 x ^ { 3 } - 5 x ^ { 2 } - 36 x - 36$
B) $x ^ { 4 } + 13 x ^ { 2 } - 4 x ^ { 3 } - 36 x + 36$
C) $x ^ { 4 } + 13 x ^ { 2 } + 4 x ^ { 3 } + 36 x + 36$
D) $4 x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - 36 x - 36$
E) $x ^ { 4 } + 4 x ^ { 3 } + 5 x - 36$