Exploring Polynomials and Multiplicity

When setting up this synthetic division to divide 2x⁴ + 3x² - x + 2 by (x - 1), what number goes into the red circle?

• -1

• 1

• 2

• cannot be determined

Polynomial function f(x) =  x⁴ - 4x³ + 5x² - 4x + 4 is known to have a zero of x = 2 with a multiplicity of 2. Does this mean (x - 2) divides the polynomial twice?

• yes

• no

(x - 2) divides polynomial function
 f(x) =  x⁴ - 4x³ + 5x² - 4x + 4 two times.  What numbers should be used to set up the second synthetic division if the goal is to factor f(x)?

•    -4   5   -4   4

•    -2   1   -2

•   -4   2   -4

• cannot be determined

(x - 2) divides polynomial function
 f(x) =  x⁴ - 4x³ + 5x² - 4x + 4 two times.  Which of these is a complete factorization of f(x)?

• (x - 2)²(x - 1)(x + 1)

• (x - 2)² (x + 1)

• (x - 2)²(x² + 1)

• cannot be determined

Find the zeros of this polynomial and describe their multiplicity.
f(x) = 3x⁴ - 12x²

• 0 multiplicity 3, -2 multiplicity 1, 2 multiplicity 1

•  -2 multiplicity 1, 2 multiplicity 1

• 0 multiplicity 2, -2 multiplicity 1, 2 multiplicity 1

• 3 multiplicity 4, -12 multiplicity 2

A polynomial factors as
f(x) = x³(x + 1)²(x - 4). List the zeros of f(x) and their multiplicity.

• 0 multiplicity 3, 1 multiplicity 2, -4 multiplicity 1

• 0 multiplicity 3, -1 multiplicity 2, 4 multiplicity 1

• 1 multiplicity 2, -4 multiplicity 1

• -1 multiplicity 2, 4 multiplicity 1

When a polynomial function f(x) is divided by
 (x - 8), the remainder is 12. 
What is f(8)?

• -8

• 8

• 12

• cannot be determined

When a polynomial function f(x) is divided by
 (x + 2), the remainder is 0. 
What is f(-2)?

• -2

• 0

• 2

• cannot be determined

If (x + 3) divides a polynomial function evenly, then which of these is known to be a zero of the polynomial?

• x = -3

• x = 3

• x = 0

• cannot be determined

When setting up this synthetic division, what polynomial is being divided?

• 2x⁴ + 3x² - x + 2

• 2x⁴ + 3x³ - x² + 2

• 2x⁴ - 3x² + x - 2

• cannot be determined