$$\text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder}$$

Question 1

Which of the following best describes the result of dividing $$6x^3$$ - $$x^2 - 5x + 4$ by 3x - 2$ using long division?

Accepted Answer

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

Question 2

When dividing $$2x^4$$ + $$3x^3 - 7x - 10$ by x^2$ - 2x$$, what is the correct quotient and remainder?

Accepted Answer

$$2x^2 + 7x + 14$ with remainder 21x - 10$

Question 3

Which of the following is the correct setup for synthetic division when dividing $$x^3$$ + $$4x^2 - 5x + 5$ by x - 3$?

Accepted Answer

Use $$c = 3$$ and coefficients $$1, 4, -5, 5$$

Question 4

What is the remainder when $$f(x) = 3x^3$ + 4x^2$ - 5x + 3 is $$divided by $$x + 4$$?

Accepted Answer

$$-105$$

Question 5

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

Accepted Answer

$$53$$

Question 6

If $$x + 2 is a $$factor of $$x^3 - 7x - 6$, what is the value of f(-2$$)$$?

Accepted Answer

$$0$$

Question 7

Which of the following statements is true about the Factor Theorem?

Accepted Answer

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

Question 8

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

Accepted Answer

$$x + 1$$

Question 9

What is the solution set for $$15x^3$$ + $$14x^2 - 3x - 2 = 0$?

Accepted Answer

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

Question 10

Which of the following best describes the structure of the division algorithm for polynomials?

Accepted Answer

$$	ext{dividend} = 	ext{divisor} 	imes 	ext{quotient} + 	ext{remainder}$$

Question 11

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

Accepted Answer

$$1, -2, -3$$

Question 12

If you divide $$x^2 + 14x + 45$ by x + 9$, what is the quotient?

Accepted Answer

$$x + 5$$

Question 13

Which of the following is a necessary step when performing long division of polynomials?

Accepted Answer

Arrange the dividend and divisor in descending order of degree.

Question 14

What is the remainder when $$x^3$$ + $$4x^2 - 5x + 5$ is divided by x - 3$?

Accepted Answer

$$53$$

Question 15

If $$f(x) = x^3$ + 4x^2$ - 5x + 5$$, what is $$f(3)$$?

Accepted Answer

$$53$$

Question 16

Which of the following is NOT a valid use of synthetic division?

Accepted Answer

Dividing by $$x^2 + 1$

Question 17

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

Accepted Answer

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

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

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