Deck 2: Polynomial and Rational Functions

Question 1

Find the domain of the rational function

A)
B)
C)
D)

Answer 1

Question 2

Find the domain of the rational function.

A)
B)
C)
D)

Answer 2

Question 3

Find the domain of the rational function
.

A)
B)
C)
D)

Answer 3

Question 4

Find the domain of the function

A)
B)
C)
D)

Question 5

Find the domain of the function

A)
B)
C)
D)

Question 6

For the rational function

, find all the vertical asymptotes and horizontal asymptotes.

A)
B)
C)
D)

Question 7

For the rational function

, find all vertical and horizontal asymptotes.

A)
B)
C)
D)

Question 8

For the rational function

, find all vertical and horizontal asymptotes.

A) vertical asymptote at x = 2, there is no horizontal asymptote
B) vertical asymptote at x = -2, horizontal asymptote at y = 2
C) vertical asymptote at x = -2, there is no horizontal asymptote
D) vertical asymptote at x = 2, horizontal asymptote at y = -2

Question 9

Professor Ito is teaching a large lecture course and is trying to learn students' names. The number of names he can remember,

, increases with each week in the semester, t, and is given by the rational function:
How many students' names does Professor Ito know by the first week of the semester? How many students' names should he know by the end of the semester (16 weeks)? Round your answer to the nearest whole number.

Question 10

Use the graphing strategy to graph the rational function.

Question 11

Match the rational function to the graph.

A)
B)
C)
D)

Question 12

Match the graph to the rational function.

A)
B)
C)
D)

Question 13

For the given graph of the rational function determine: (a) all intercepts, (b) all asymptotes, and (c) equation of the rational function.

Question 14

For the given graph of the rational function determine: (a) all intercepts, (b) all asymptotes, and (c) equation of the rational function.

A)
B)
C)
D)

Question 15

For the rational function

, find the equation of the slant asymptote.

Question 16

For the rational function

, find the equation of the slant asymptote.

Question 17

Use the graphing strategy to graph the rational function.

Question 18

Use the graphing strategy to graph the rational function.

Question 19

Use the graphing strategy to graph the rational function.

Question 20

Find all zeros (real and complex). Factor the polynomial as a product of linear factors.

Question 21

Find all zeros (real and complex). Factor the polynomial as a product of linear factors.

Question 22

Find all zeros (real and complex). Factor the polynomial as a product of linear factors.

Question 23

Given a zero of the polynomial, determine all other zeros (real and complex) and write the polynomial in terms of a product of linear factors.

Question 24

Given a zero of the polynomial, determine all other zeros (real and complex) and write the polynomial in terms of a product of linear factors.

A)


B)


C)


D)

Question 25

Factor the polynomial as a product of linear factors.

Question 26

Factor the polynomial as a product of linear factors.

A)

B)

C)

D)

Question 27

Factor the polynomial as a product of linear factors.

Question 28

Factor the polynomial as a product of linear factors.

A)

B)

C)

D)

Question 29

Find a polynomial of minimum degree that has these zeros.
0, 3 + 3i, 3 - 3i

Question 30

Find a polynomial of minimum degree that has these zeros.
2, 5 + 7i, 5 - 7i

Question 31

Find a polynomial of minimum degree that has these zeros.
1, 1 + 3i, 1 - 3i, 3i, -3i

Question 32

Determine whether the number -7 is a zero of
. If it is, find the other real zeros.

A) -7 is not a zero.
B) -7 is a zero and the others are -4 and 7.
C) -7 is a zero and the others are 4 and -7.
D) -7 is a zero and there are no other real zeros.

Question 33

Determine whether the number -4 is a zero of
. If it is, find the other real zeros.

A) -4 is not a zero
B) -4 is a zero and the others are 2 and -29
C) -4 is a zero and the other is 42.
D) -4 is a zero and there are no other real zeros.

Question 34

Determine whether -9 is a zero of the polynomial. If it is, then find the other real zeros. P(x) = x³ + 8x² - 8x + 9

A) -9 is not a zero
B) -9 is a zero and the other zeros are 8 and -8
C) -9 is a zero and the other is 9
D) -9 is a zero and there are no other real zeros.

Question 35

Given that 6 is a zero of the polynomial P(x) = x³ - 18x² + 107x - 210, determine all other zeros.

A) -5 and 7
B) 5 and 7
C) -5 and -7
D) 5 and -7

Question 36

Use Descartes' rule of signs to determine the possible number of positive real zeros and negative real zeros.
.

A)
B)
C)
D)

Question 37

Use Descartes' Rule of signs to determine the possible number of positive real zeros and negative real zeros. P(x) = 2x⁵ + 4x⁴ - 3x - 6

A)
B)
C)
D)

Question 38

Use Descarte's Rule of signs to determine the possible number of positive real zeros and negative real zeros. P(x) = 5x³ + 2x² + 4x - 9

A)
B)
C)
D)

Question 39

Use Descartes' rule of signs to determine the possible number of positive real zeros, negative real zeros, and imaginary zeros. P(x) = x³ - 10x² + 8x + 6

A)
B)
C)
D)

Question 40

Use the rational zero theorem to list the possible rational zeros of the polynomial

A)
B)
C)
D)

Question 41

Use the rational zero theorem to list the possible rational zeros.
P(x) = 5x³ + 2x² - 2x + 1

A)
B)
C)
D)

Question 42

Use the rational zero theorem to list the possible rational zeros.
P(x) = 7x³ - 4x² + 3x - 21

A)
B)
C)
D)

Question 43

Given that -1 is a zero of the polynomial P(x) = x³ + 10x² + 29x + 20, determine all other zeros and write the polynomial in terms of a product of linear factors.

Question 44

Use the rational zero theorem to list the possible rational zeros.
P(x) = x³ + 8x² + 8x + 77

Question 45

Use the intermediate value theorem to approximate the real zero in the indicated interval. Approximate to two decimal places.

Question 46

Use the rational zero theorem to list the possible rational zeros.
P(x) = 4x³ - 7x² + 12x - 91

Question 47

Use Descartes' rule of signs to determine the possible number of positive real zeros, negative real zeros, and imaginary zeros.
P(x) = x³ - 3x² + 8x + 9

Question 48

Use Descartes' rule of signs along with the rational root theorem to sketch a graph of the polonomial.
P(x) = x³ - 4x² - 6x + 14

Question 49

For the polynomial

, use synthetic division to find

.

Question 50

For the polynomial function f(x) = 4x³ + 5x - 7, use synthetic division to find f(1).

Question 51

Use long division to divide the polynomials. Express the answers in the form of

and

A)
B)
C)
D)

Question 52

Use long division to divide 9x³ + 36x² + 57x + 42 by 3x + 6.

Question 53

Use long division to divide 12x⁴ + 44x³ + 30x² + 28x + 30 by 2x + 6.

Question 54

Divide the polynomials by the linear factor with synthetic division.

Question 55

Divide the polynomials by either long division or synthetic division.

Question 56

Use synthetic division to divide the polynomials. Express the answers in the form of

And

A)
B)
C)
D)

Question 57

Use synthetic division to divide 6x³ + 12x² + 24x + 18 by 3x + 3.

Question 58

Use synthetic division to divide 12x⁴ + 13x³ + 18x² + 11x + 2 by 3x + 1.

Question 59

Divide the polynomials using long division.

Question 60

Divide the polynomials using long division.

Question 61

Divide the polynomials using long division.

Question 62

Divide the polynomials using synthetic division.

Question 63

Divide the polynomials using long division.

Question 64

Divide the polynomials using synthetic division.

Question 65

Divide the polynomials using long division.

Question 66

Divide the polynomials using synthetic division.

Question 67

Divide the polynomials using long division.

Question 68

Divide the polynomials using synthetic division.

Question 69

Determine if the function

Is a polynomial. If it is, state the degree.

A) Not a polynomial
B) a polynomial of degree -22
C) a polynomial of degree 49
D) a polynomial of degree 4

Question 70

Determine if the function

Is a polynomial. If it is, state the degree.

A) Not a polynomial
B) a polynomial of degree 34
C) a polynomial of degree 32
D) a polynomial of degree -32

Question 71

Determine if the function

Is a polynomial. If it is, state the degree.

A) Not a polynomial
B) a polynomial of degree 31
C) a polynomial of degree 22
D) a polynomial of degree 51

Question 72

Find all the real zeros (and state their multiplicity) of the polynomial function.

A) 0, -2, 3
B) -2, 3
C)
D)

Question 73

Find all the real zeros (and state their multiplicity) of the polynomial function.

A) 0, 16 (multiplicity 2)
B) 0, -16, (multiplicity 2)
C) 1, 16, (multiplicity 2), 18 (multiplicity 2)
D)

Question 74

Find all the real zeros (and state their multiplicity) of the polynomial function.
Y = x⁶ - 18x⁵ + 81x⁴

A) 0, 9
B) 0 (multiplicity 4), 9 (multiplicity 2)
C) 9 (multiplicity 2)
D) 0 (multiplicity 6), 9 (multiplicity 2)

Question 75

Find a polynomial of minimum degree with zeros 5, 4 and -5 (of multiplicity 2)

A)
B)
C)
D)

Question 76

Find a polynomial of minimum degree that has zeros -4, 0 (multiplicity 3) and 1.

A) y = x⁵ + 3x⁴ - 4x³
B) y = x² - 3x - 4
C) y = x⁵ - 3x⁵ - 4x³
D) y = x² + 3x - 4x

Question 77

Find a polynomial of minimum degree with zeros 0,
, and

.

A)
B)
C)
D)

Question 78

Find a polynomial of minimum degree that has the zeros

(with multiplicity 2) and

(with multiplicity 2).

A) f(x) = x⁴ + 6x² + 9
B) f(x) = x⁴ - 6x² + 9
C) f(x) = x⁴ + 9
D) f(x) = x² + 6x + 9

Question 79

For the polynomial function

, determine whether the graph touches or crosses at the x - intercept (-20,0).

A) crosses the y - axis at (-20,0)
B) touches the y - axis at (-20,0)
C) crosses the x - axis at (-20,0)
D) touches the x - axis at (-20,0)

Question 80

For the polynomial function f(x) = x⁷(x + 14)²(x - 8), determine whether the graph touches or crosses at the x-intercept (0,0).

A) touches the x-axis at (0,0)
B) crosses the x-axis at (0,0)
C) touches the x-axis at (8,0)
D) neither