Question 1

What is the result of dividing $$2x^3$$ - $$3x^2 - 11x + 7$ by x - 3$ using long division?

Accepted Answer

$$2x^2 + 3x - 2$$ + \frac{1}{x-3}$$

Question 2

When using synthetic division to divide $$x^3 - 7x - 6$ by x + 2$, what is the quotient?

Accepted Answer

$$x^2 - 2x - 3$

Question 3

According to the Remainder Theorem, what is the remainder when $$f(x) = 3x^3$ + 4x^2$ - 5x + 3 is $$divided by $$x + 4$$?

Accepted Answer

$$-105$$

Question 4

If $$f(x) = 15x^3$ + 14x^2$ - 3x - 2$$ and $$-1 is a $$zero of $$f(x)$$, which of the following is a factor of $$f(x)$$?

Accepted Answer

$$x + 1$$

Question 5

What is the solution set for $$15x^3$$ + $$14x^2 - 3x - 2 = 0$ given that -1$ is a zero?

Accepted Answer

$$\left\{-1,\ -\frac{1}{3},\ \frac{2}{5}\right\}$$

Question 6

Which statement best describes the Factor Theorem?

Accepted Answer

If $$f(c) = 0$$, then $$x - c is a $$factor of $$f(x).$$

Question 7

What is the degree of the remainder when dividing a polynomial $$f(x) by$$ $$d(x)$$ using the Division Algorithm?

Accepted Answer

Less than the degree of $$d(x)$$

Question 8

In synthetic division, what must the divisor be written as?

Accepted Answer

$$x - c$$

Question 9

If the remainder when dividing $$f(x) by$$ $$x - c is $$zero, what does this imply?

Accepted Answer

$$x - c is a $$factor of $$f(x)$$

Question 10

A chemist models the concentration of a reactant over time with $$f(x) = 2x^3$ - 3x^2$ - 11x + 7. If $$she wants to know the concentration at $$x = 3$$, which theorem can she use?

Accepted Answer

Remainder Theorem

Question 11

Which step is NOT part of the long division process for polynomials?

Accepted Answer

Add the product to the dividend.

Question 12

A medical researcher uses $$f(x) = 3x^3$ + 4x^2$ - 5x + 3 to $$model a patient's response. What is $$f(-4)$$?

Accepted Answer

$$-105$$

Question 13

If $$f(x) is $$divided by $$x - c$$ and the remainder is $$r$$, what is $$r$$ according to the Remainder Theorem?

Accepted Answer

$$f(c)$$

Question 14

Which of the following is a necessary condition for using synthetic division?

Accepted Answer

The divisor must be linear and in the form $$x - c$$

Question 15

If $$f(x) = x^3$ - 7x - 6$$ and $$x + 2 is a $$factor, what is $$f(-2)$$?

Accepted Answer

$$0$$

Question 16

A business models profit with $$f(x) = 2x^3$ - 3x^2$ - 11x + 7. If$$ $$x = 3 is a $$solution to $$f(x) = 0$$, what does this mean?

Accepted Answer

$$x - 3 is a $$factor of $$f(x)$$

Question 17

What is the quotient when $$15x^3$$ + $$14x^2 - 3x - 2$ is divided by x + 1$?

Accepted Answer

$$15x^2 - x - 2$

Question 18

Which of the following best describes the Division Algorithm for polynomials?

Accepted Answer

$$f(x) = d(x)q(x) + r(x)$$, where $$r(x) is $$zero or has degree less than $$d(x)$$

Dividing Polynomials: Long Division, Synthetic Division, Remainder and Factor Theorems

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