Deck 3: Polynomial and Rational Functions

Question 1

Complete the square to write the function in shifted form. f(x) = x² + 7x + 1

A) f(x) =
B) f(x) =
C) f(x) =
D) f(x) =

Answer 1

f(x) =

Question 2

Use the following to answer questions :
An automobile manufacturer can produce up to 300 cars per day. The profit made from the sale of these vehicles can be modeled by the function , where P(x) is the profit in dollars and x is the number of automobiles made and sold.
Find the y-intercept and explain what it means in this context.

Answer 2

(0, -22,000); when no cars are produced, there is a loss of $22,000

Question 3

Use synthetic division and the remainder theorem to evaluate P(-3) if P(x) = 2x³ - 3x² + 5x - 13.

A) -110
B) -109
C) -108
D) -107

Answer 3

-109

Question 4

Use the following to answer questions :
An automobile manufacturer can produce up to 300 cars per day. The profit made from the sale of these vehicles can be modeled by the function , where P(x) is the profit in dollars and x is the number of automobiles made and sold.
What is the maximum profit?

A) $98,000
B) $99,000
C) $100,000
D) $101,000

Question 5

Use synthetic division and the remainder theorem to show that x = -2 is a zero of P(x) = x³ - 4x² - 4x + 16.

Question 6

Find the vertex.
f(x) = x² + 10x - 8

Question 7

Use the following to answer questions :
An automobile manufacturer can produce up to 300 cars per day. The profit made from the sale of these vehicles can be modeled by the function , where P(x) is the profit in dollars and x is the number of automobiles made and sold.
How many cars should be made and sold to maximize profit?

A) 25
B) 75
C) 150
D) 1500

Question 8

Graph the function using end behavior, intercepts and by completing the square to write the function in shifted form. Clearly state the transformations used to obtain the graph, and label the vertex and all intercepts (if they exist). Use the quadratic formula to find the x-intercepts.
f(x) = 2x² - 9x + 4

Question 9

Find the intercepts. Round to the nearest tenth if necessary.
f(x) = 3x² + 8x - 6

Question 10

Find the intercepts.
f(x) = x² - 10x + 24

Question 11

Use the vertex/intercept formula, , to find zeroes, real or complex of the formula. f(x) = 3(x + 1)² - 2

A)
B)
C)
D)

Question 12

Graph the function using end behavior, intercepts and by completing the square to write the function in shifted form. Clearly state the transformations used to obtain the graph, and label the vertex and all intercepts (if they exist). Use the quadratic formula to find the x-intercepts. f(x) = 3x² + 5x + 1

A) left 1, down 2, stretched vertically

B) right 1, down 2, stretched vertically

C) left .8, down 1.1, stretched vertically

D) right .8, down 1.1, stretched vertically

Question 13

Complete the square to write the function in shifted form.
f(x) = x² + 6x + 1

Question 14

Determine the equation of the function shown.

A) f(x) = -x² + 1
B) f(x) = -x² - 1
C) f(x) = -(x + 1)²
D) f(x) = -(x - 1)²

Question 15

Graph the function using end behavior, intercepts and by completing the square to write the function in shifted form. Clearly state the transformations used to obtain the graph, and label the vertex and all intercepts (if they exist). Use the quadratic formula to find the x-intercepts.
f(x) = -x² + 4x - 3

Question 16

Graph the function using the concavity, y-intercept, x-intercept(s), vertex, and symmetry. f(x) =

A)
(Gridlines are spaced one unit apart.)
B)
(Gridlines are spaced one unit apart.)
C)
(Gridlines are spaced one unit apart.)
D)
(Gridlines are spaced one unit apart.)

Question 17

Use synthetic division and the remainder theorem to evaluate P(4) if P(x) = x³ - 5x² - 4x + 20. Verify using a second method.

Question 18

Use the following to answer questions :
An automobile manufacturer can produce up to 300 cars per day. The profit made from the sale of these vehicles can be modeled by the function , where P(x) is the profit in dollars and x is the number of automobiles made and sold.
Find the x-intercepts and explain what they mean in this context.

Question 19

Graph the function using the end behavior, y-intercept, x-intercept(s), vertex, and symmetry. Label the vertex and all intercepts (if they exist). (Round to tenths as necessary).
f(x) = -x² + 2x + 5

Question 20

Graph the function using end behavior, intercepts and by completing the square to write the function in shifted form. Clearly state the transformations used to obtain the graph, and label the vertex and all intercepts (if they exist). Use the quadratic formula to find the x-intercepts.
f(x) = 2x² + 9x + 4

Question 21

A polynomial P of degree 3 with integer coefficients has zeroes x = 5, , and . Use the factor theorem to write the polynomial in factored form and standard form.

Question 22

Use the factor theorem to find the polynomial of degree 4 having zeroes , , x = 3 + i, x = 3 - i. Assume a lead coefficient of 1.

Question 23

Find a cubic polynomial with real coefficients having roots x = -2 and x = 1 + 4i. Assume a lead coefficient of 1.

A) P(x) = x³ + 13x + 34
B) P(x) = x³ + 13x - 34
C) P(x) = x³ + 4x² + 21x + 34
D) P(x) = x³ - 4x² + 21x - 34

Question 24

Use the rational roots theorem to write the function in factored form and find all zeroes. Note a = 1. p(x) = x⁴ + 6x³ - 13x² - 66x + 72

A) p(x) = (x - 1)(x - 3)(x + 4)(x + 6); zeroes: 1, 3, -4, -6
B) p(x) = (x + 1)(x + 3)(x - 4)(x - 6); zeroes: -1, -3, 4, 6
C) p(x) = (x - 1)(x + 3)(x - 4)(x + 6); zeroes: 1, -3, 4, -6
D) p(x) = (x + 1)(x - 3)(x + 4)(x - 6); zeroes: -1, 3, -4, 6

Question 25

List all possible rational roots for the polynomial but do not solve.
f(x) = 21x³ - 5x² + x - 175

Question 26

Use the following to answer questions :
p(x) = x⁴ - 5x³ + 5x² + 5x - 6
Use the rational roots theorem to list all possible rational roots.

A)
B)
C)
D)

Question 27

Use the rational roots theorem to write the function in factored form and find all zeroes. Note a = 1.
p(x) = x³ + 4x² - 9x - 36

Question 28

Find a quartic polynomial (degree 4) with real coefficients having roots x = -5i and x = i. Assume a lead coefficient of 1.

A) P(x) = x⁴ + 32x³ + 25x² + 7x + 175
B) P(x) = x⁴ + 25x³ + 32x² + 7x + 175
C) P(x) = x⁴ + 32x² + 175
D) P(x) = x⁴ + 25x² + 175

Question 29

Use synthetic division and the remainder theorem to show x = 1 - 3i is a zero of P(x) = -x³ - x² - 4x - 30.

Question 30

Find a polynomial P with real coefficients having degree 5, only one real, rational root, and zeroes x = 2 and x = 1 + 5i. Assume a lead coefficient of 1. Leave your answer in factored form.

Question 31

Find the zeroes of the polynomial using any combination of the rational roots theorem, synthetic division, testing for 1 and -1, and/or the remainder and factor theorems.
p(x) = 4x⁴ - 3x³ - 30x² - x + 30

Question 32

Find all rational zeroes of the function given and use them to write the function in factored form. Use the factored form to state all zeroes of the function. Begin by applying the tests for 1 and -1. p(x) = 4x⁴ + x³ + 33x² +9x - 27

A) p(x) = (x - 1)(4x - 3)(x - 3i)(x + 3i); zeroes: 1, , 3i, -3i
B) p(x) = (x - 1)(4x + 3)(x - 3i)(x + 3i); zeroes: 1, - , 3i, -3i
C) p(x) = (x + 1)(4x - 3)(x - 3i)(x + 3i); zeroes: -1, , 3i, -3i
D) p(x) = (x + 1)(4x + 3)(x - 3i)(x + 3i); zeroes: -1, - , 3i, -3i

Question 33

Find a cubic polynomial with real coefficients having roots x = -3 and x = -i. Assume a lead coefficient of 1.

Question 34

Use the remainder theorem to show x = 2i is a zero of P(x) = x³ + 2x² + 4x + 8.

Question 35

Factor completely. Then state the multiplicity of the roots and the degree of P.
P(x) = x³ - 2x² - 15x + 36.

Question 36

A polynomial P of degree 3 with integer coefficients has zeroes x = 3, x = -9, and x = 1. Use the factor theorem to write the polynomial in factored form.

A) P(x) = (x + 3)(x - 9)(x + 1)
B) P(x) = (x - 3)(x + 9)(x - 1)
C) P(x) = (3x + 1)(-9x + 1)(x + 1)
D) P(x) = (-3x + 1)(9x + 1)(-x + 1)

Question 37

Find a quartic polynomial (degree 4) with real coefficients having roots x = and x = 1 + i. Assume a lead coefficient of 1.

Question 38

Find the zeroes of the polynomial using any combination of the rational roots theorem, synthetic division, testing for 1 and -1, and/or the remainder and factor theorems. p(x) = x⁴ + 3x³ + 3x² - 17x - 18

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

Question 39

Use the factor theorem to find the polynomial of degree 3 having zeroes x = 1, x = 1 - 4i, and x = 1 + 4i. Assume a lead coefficient of 1.

A) P(x) = x³ - 3x² + 20x - 16
B) P(x) = x³ - 3x² + 20x - 17
C) P(x) = x³ - 3x² + 19x - 16
D) P(x) = x³ - 3x² + 19x - 17

Question 40

Use the intermediate value theorem to verify that the polynomial f(x) = -2x³ - 4x² + 3x - 4 has at least one zero "c_i" in the interval [-3, -1]. Do not find the zeroes.

A) Yes
B) No

Question 41

Find all zeroes (real and complex) of the polynomial.
p(x) = 4x³ - 5x² - 23x + 6

Question 42

Gather information on the polynomial using the rational roots theorem, testing for 1 and -1, Descartes's rule of signs, and the upper and lower bounds property. Respond explicitly to each.
p(x) = x⁴ - 3x³ + 9x - 27

Question 43

State the degree and end behavior of the function. f(x) = (x - 1)⁶(x + 3)(x - 3)³

A) degree 9, up/up
B) degree 9, down/up
C) degree 10, up/up
D) degree 10, down/up

Question 44

State the end behavior and y-intercept of the function. Do not graph.
f(x) = x⁵ + 5x⁴ - 5x² - 3x + 5

Question 45

Use the following to answer questions : (Gridlines are spaced one unit apart.)
Find the minimum possible degree of the function and write it in factored form. Assume all zeroes are real.

A) f(x) = -(x + 1)(x - 2)
B) f(x) = -(x + 1)²(x - 2)
C) f(x) = (x + 1)(x - 2)²
D) f(x) = (x + 1)²(x - 2)

Question 46

Use the following to answer questions : (Gridlines are spaced one unit apart.)
State whether the degree of the function is even or odd.

A) even
B) odd

Question 47

Use the following to answer questions : (Gridlines are spaced one unit apart.)
Use the graph to estimate the zeroes of the function, then state whether their multiplicity is even or odd.

A) -1, even; 2 odd
B) -1, odd; 2 even
C) -1, even; 2 even
D) -1, odd; 2 odd

Question 48

Use the Guidelines for Graphing Polynomial Functions to graph the polynomial.
f(x) = x³ - 3x - 2

Question 49

For the complex polynomial below, one of its zeroes is z = 5i. Use the given zero to help find all zeroes of the polynomial, then write the polynomial in completely factored form. Hint: synthetic division and the quadratic formula can be applied to all polynomials, even those with complex coefficients.
C(z) = z³ + (1 - 5i)z² + (-6 - 5i)z + 30i.

Question 50

State the degree, end behavior, and y-intercept of the function.
f(x) = -(x - 5)(x - 1)⁸(x - 2)²

Question 51

State the end behavior of the function. f(x) = 5x⁴ + 2x³ - 5x² - 5x - 3

A) down/up
B) up/down
C) down/down
D) up/up

Question 52

Use the following to answer questions :
p(x) = x⁴ - 5x³ + 5x² + 5x - 6
Use the information gathered, testing for 1 and -1, synthetic division, and the factor theorem to factor p completely.

A) p(x) = (x + 1)(x - 1)(x + 2)(x + 3)
B) p(x) = (x + 1)(x - 1)(x - 2)(x + 3)
C) p(x) = (x + 1)(x - 1)(x + 2)(x - 3)
D) p(x) = (x + 1)(x - 1)(x - 2)(x - 3)

Question 53

Find all zeroes (real and complex) of the polynomial.
p(x) = 6x³ + 29x² - 149x + 24

Question 54

Use the Guidelines for Graphing Polynomial Functions to graph the polynomial. f(x) = x³ - 4x² + x + 6

A)

B)

C)

D)

Question 55

State the end behavior of the function. f(x) = -3x⁵ + 5x⁴ + x² - 2x + 2

A) down/up
B) up/down
C) down/down
D) up/up

Question 56

State the end behavior and y-intercept of the function. Do not graph.
f(x) = -2x⁴ - 2x³ + x² - 2x + 5

Question 57

Use the following to answer questions :
p(x) = x⁴ - 5x³ + 5x² + 5x - 6
Apply Descartes's rule of signs to count the possible number of positive, negative, and complex roots (organize a table).

A)
B)
C)
D)

Question 58

Sketch the graph of the function using the degree, end behavior, x- and y-intercepts, zeroes of multiplicity, and a few midinterval points to round out the graph. Connect all points with a smooth, continuous curve.
f(x) = (x - 1)³(x + 2)(x + 3)

Question 59

Use Descartes's rule of signs to determine the possible combinations of real and complex roots for the polynomial. Then graph the function on the standard window of a graphing calculator and adjust it as needed until you're certain all real roots are in clear view. Use this screen and a list of the possible rational zeroes (rational roots theorem) to factor the polynomial and find all zeroes (real and complex). p(x) = 20x³ - 153x² - 62x + 48

A) p(x) = (x - 8)(2x + 1)(10x + 6)
B) p(x) = (x + 8)(10x - 1)(2x + 6)
C) p(x) = (x - 8)(5x - 2)(4x + 3)
D) p(x) = (x + 8)(4x + 1)(5x + 6)

Question 60

For the complex polynomial below, one of its zeroes is x = 3 - i. Use the given zero to help find all zeroes of the polynomial, then write the polynomial in completely factored form. Hint: synthetic division and the quadratic formula can be applied to all polynomials, even those with complex coefficients. C(x) = x³ - 3x² + (3 + 3i)x + (-6 + 2i)

A) C(x) = (x - 3 - i)(x + 2i)(x + i); x = -2i, x = -i
B) C(x) = (x - 3 + i)(x - 2i)(x - i); x = 2i, x = i
C) C(x) = (x - 3 + i)(x + 2i)(x - i); x = -2i, x = i
D) C(x) = (x - 3 + i)(x - 2i)(x + i); x = 2i, x = -i

Question 61

Graph the polynomial function.
f(x) = x³ + 2x² - 5x - 6

Question 62

Use the following to answer questions : (Gridlines are spaced one unit apart.)
Give the equation related to the graph. Assume |a| = 1.

A)
B)
C)
D)

Question 63

Determine the location of the horizontal asymptote if it exists. Then determine whether the graph will cross the asymptote, and if so, where it crosses.

Question 64

Use the following to answer questions : (Gridlines are spaced one unit apart.)
State the equations of the horizontal and vertical asymptotes.

A) y = 4, x = -3
B) y = -4, x = 3
C) y = -3, x = 4
D) y = 3, x = -4

Question 65

Use the following to answer questions : (Gridlines are spaced one unit apart.)
State the equations of the horizontal and vertical asymptotes.

A) y = -4, x = 2
B) y = 4, x = -2
C) y = 2, x = -4
D) y = -2, x = 4

Question 66

Give the location of the vertical asymptote(s) if they exist, and state whether function values will change sign (positive to negative or negative to positive) from one side of the asymptote to the other.

A) x = 1, no
B) x = 1, yes
C) x = -4, no; x = 1, no
D) x = -4, yes; x = 1, yes

Question 67

Use the following to answer questions : (Gridlines are spaced one unit apart.)

-State the range of the function.

A) y $\in$ (-?, -2) $\cup$ (-2, ?)
B) y $\in$ (-?, 2) $\cup$ (2, ?)
C) y $\in$ (-?, -3)
D) y $\in$ (-?, 3)

Question 68

Use polynomial division or synthetic division to rewrite the function, then sketch the graph using transformations of a parent function and the x- and y-intercepts (if they exist).

Question 69

Graph the polynomial function.
f(x) = x⁴ + 3x³ - 3x² - 11x - 6

Question 70

Use the following to answer questions : (Gridlines are spaced one unit apart.)
Give the equation related to the graph. Assume |a| = 1.

A)
B)
C)
D)

Question 71

Use the following to answer questions : (Gridlines are spaced one unit apart.)
Use the direction/approach notation to describe the end behavior of the graph. (This should produce four statements.)

Question 72

Determine the location of the horizontal asymptote if it exists. Then determine whether the graph will cross the asymptote, and if so, where it crosses.

A) y = 0; crosses at (1, 0)
B) y = 0; does not cross
C) y = 2; does not cross
D) No horizontal asymptote

Question 73

Use the following to answer questions : (Gridlines are spaced one unit apart.)

-State the domain of the function.

A) x $\in$ (-?, -2) $\cup$ (-2, ?)
B) x $\in$ (-?, 2) $\cup$ (2, ?)
C) x $\in$ (-?, -3) $\cup$ (-3, ?)
D) x $\in$ (-?, 3) $\cup$ (3, ?)

Question 74

Give the location of the vertical asymptote(s) if they exist, and state the function's domain in set notation.

A) x = 1; {x | x $\in$ R, x ? 1}
B) x = -3; {x | x $\in$ R, x ? -3}
C) x = 4; {x | x $\in$ R, x ? 4}
D) No vertical asymptote; {x | x $\in$ R}

Question 75

Give the location of the vertical asymptote(s) if they exist, and state whether function values will change sign (positive to negative or negative to positive) from one side of the asymptote to the other.

A) x = 4, no
B) x = 4, yes
C) x = -1, no; x = 2, no
D) x = -1, yes; x = 2, yes

Question 76

Use the following to answer questions : (Gridlines are spaced one unit apart.)
Use the direction/approach notation to describe the end behavior of the graph. (This should produce four statements.)

Question 77

Determine the location of the x- and y-intercepts (if they exist), and state the behavior of the function (bounce or cut) at each x-intercept.

A) (0, 0), bounce; (-2, 0), cut
B) (0, 0), cut; (-2, 0), bounce
C) (0, 0), bounce; (-2, 0), cut; (5, 0), bounce
D) (0, 0), cut; (-2, 0), bounce; (5, 0), cut

Question 78

Use the following to answer questions : (Gridlines are spaced one unit apart.)

-State the domain of the function.

A) x $\in$ (-?, -4) $\cup$ (-4, ?)
B) x $\in$ (-?, 4) $\cup$ (4, ?)
C) x $\in$ (-?, -2) $\cup$ (-2, ?)
D) x $\in$ (-?, 2) $\cup$ (2, ?)

Question 79

Give the location of the vertical asymptote(s) if they exist, and state the function's domain in set notation.

A) x = 2; {x | x $\in$ R, x ? 2}
B) x = ; {x | x $\in$ R, x ?
}
C) x = -5, x = -1; {x | x $\in$ R, x ? -5, x ? -1}
D) No vertical asymptote; {x | x $\in$ R}

Question 80

Use the following to answer questions : (Gridlines are spaced one unit apart.)

-State the range of the function.

A) y $\in$ (-?, -4) $\cup$ (-4, ?)
B) y $\in$ (-?, 4) $\cup$ (4, ?)
C) y $\in$ (-?, -2) $\cup$ (-2, ?)
D) y $\in$ (-?, 2) $\cup$ (2, ?)