Deck 13: Roots of Polynomial Equations

Question 1

Express in the form . Find a quadratic function that has a maximum value of 2 and that has - 2 and 4 as zeros.

A)
B)
C)
D)

Answer 1

Question 2

Determine the constants (denoted with capital letters) so that the equation is an identity.

A)
B)
C)
D)
E)

Answer 2

Question 3

Find a quadratic equation with rational coefficients, one of whose roots is the given number. Write your answer so that the coefficient of is 1.

A)
B)
C)
D)
E)

Answer 3

Question 4

List the distinct roots of the following equation. For both repeated and single roots, specify their multiplicity. Enter , ... where etc. are the roots of the polynomial and is the multiplicity of is the multiplicity of etc.

A) (3, 1), (2, 4), (9, 1)
B) (3, 1), (2, 1), (9, 1)
C) (3, 1), (2, 5), (9, 1)
D) (3, 1), (2, 6), (9, 1)
E) (3, 4), (2, 4), (9, 4)

Question 5

Use long division to find the quotient and the remainder.

A)
B)
C)
D)
E)

Question 6

Find the rational roots and solve the equation. 4x ³ + x ² - 20x - 5 = 0

A)
B)
C)
D)
E)

Question 7

Use synthetic division to find the quotient and the remainder.

A)
B)
C)
D)
E)

Question 8

Determine the constants (denoted by capital letters) so that the equation is an identity.

A)
B)
C)
D)
E)

Question 9

Use Descartes's rule of signs to obtain information regarding the roots of the equation.

A) 1 negative root, 8 complex roots
B) 9 complex roots
C) 1 negative root, 9 complex roots
D) 1 positive root, 1 negative real root, 7 complex roots
E) 1 positive root, 8 complex roots

Question 10

Find the rational roots and solve the equation. 3x ³ - 19x ² + 21x - 5 = 0

A)
B)
C)
D)
E)

Question 11

Use synthetic division to find the quotient and the remainder.

A)
B)
C)
D)
E)

Question 12

Use a graph to determine whether the equation has at least one real root.

A) three real roots
B) two real roots
C) no real roots
D) one real root
E) It is not possible to make this determination by graphing the equation.

Question 13

Use long division to find the quotient and the remainder.

A)
B)
C)
D)
E)

Question 14

Find the rational roots and solve the equation. 4x ³ - 10x ² - 25x + 4 = 0

A)
B)
C)
D)
E)

Question 15

Determine which of the following real numbers is a root of the equation.

A)
B)
C)
D)
E)

Question 16

Determine which of the following real numbers is a root of the equation.

A)
B)
C)
D)
E)

Question 17

One root of the equation is given. Determine the remaining roots.

A)
B)
C)
D)
E)

Question 18

The equation has exactly one positive root. Locate the root between successive hundredths. x ³ - 3x ² + 3x - 12 = 0

A) between 3.11 and 3.12
B) between 4.31 and 4.32
C) between 2.22 and 2.23
D) between 3.22 and 3.23
E) between 2.02 and 2.03

Question 19

Use long division to find the quotient and the remainder.

A)
B)
C)
D)
E)

Question 20

Determine upper and lower bounds for the real roots of the equation. 5x ⁴ - 10x - 12 = 0

A) 2 is an upper bound, - 1 is a lower bound
B) 8 is an upper bound, - 1 is a lower bound
C) 1 is an upper bound, - 2 is a lower bound
D) 2 is an upper bound, - 4 is a lower bound
E) 8 is an upper bound, - 4 is a lower bound

Question 21

Determine the constants (denoted by capital letters) so that the equation is an identity.

A)
B)
C)
D)
E)

Question 22

Factor the denominator of the given rational expression, and then find the partial fraction decomposition.

A)
B)
C)
D)
E)

Question 23

Factor the denominator of the given rational expression, and then find the partial fraction decomposition.

A)
B)
C)
D)
E)

Question 24

Determine which of the following polynomials is not reducible.

A)
B)
C)
D)
E)

Question 25

Factor the denominator of the given rational expression, and then find the partial fraction decomposition.

A)
B)
C)
D)
E)