Question 1

Which of the following statements best describes when a polynomial $d(x)$ is a factor of another polynomial $P(x)$?

Accepted Answer

When the remainder from dividing $P(x)$ by $d(x)$ is zero.

Question 2

Given $P(x) = x^3 + 2x^2 - 5x - 6$, which of the following is a factor of $P(x)$?

Accepted Answer

$x - 3$

Question 3

When dividing a polynomial $P(x)$ by $d(x)$, which of the following correctly expresses the division algorithm?

Accepted Answer

$P(x) = d(x)Q(x) + R(x)$

Question 4

If $P(x)$ is divided by $d(x)$ and the remainder is $R(x)$, what must be true about the degree of $R(x)$?

Accepted Answer

The degree of $R(x)$$ must be less than the degree of $d(x)$.

Question 5

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

Accepted Answer

$f(2) = 2^3 - 4(2^2) + 5(2) - 2 = 8 - 16 + 10 - 2 = 0$

Question 6

What is the name of the procedure used to divide polynomials by linear divisors of the form $x - c$?

Accepted Answer

Synthetic division

Question 7

Perform synthetic division of $P(x) = 2x^2 + 7x - 5$ by $x + 3$. What is the remainder?

Accepted Answer

$P(-3) = 2(-3)^2 + 7(-3) - 5 = 18 - 21 - 5 = -8$

Question 8

If $g(x) = x^2 - 4x + 7$, what is $g(3)$?

Accepted Answer

$g(3) = 3^2 - 4(3) + 7 = 9 - 12 + 7 = 4$

Question 9

If $P(x) = x^3 - 7x^2 + 16x - 12$ and $x = 3$ is a solution to $P(x) = 0$, which of the following is another solution?

Accepted Answer

$x = 4$

Question 10

A chemist is modeling the concentration of a substance with $C(x) = x^3 - 13x^2 + 55x - 75$. If $x = 3$ is a root, which of the following is also a root?

Accepted Answer

$x = 5$

Question 11

Which of the following is the correct application of the Remainder Theorem?

Accepted Answer

The remainder when $f(x)$ is divided by $x - c$ is $f(c)$.

Question 12

A company models its profit with $P(x) = x^3 - 4x^2 + 5x - 2$. If $x = 1$ is a root, what is the profit when $x = 1$?

Accepted Answer

$P(1) = 1^3 - 4(1^2) + 5(1) - 2 = 1 - 4 + 5 - 2 = 0$

Question 13

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

Accepted Answer

Dividing by a quadratic divisor

Question 14

If $P(x) = x^3 + 2x^2 - 5x - 6$ and $P(-1) = 0$, what does this imply?

Accepted Answer

$x + 1$ is a factor of $P(x)$

Question 15

A medical researcher models the spread of a virus with $V(x) = x^3 - 7x^2 + 16x - 12$. If $x = 3$ is a solution, what is the value of $V(3)$?

Accepted Answer

$V(3) = 0$

Question 16

Which of the following best describes the degree of the quotient $Q(x)$ when dividing $P(x)$ by $d(x)$?

Accepted Answer

The degree of $Q(x)$ is the degree of $P(x)$ minus the degree of $d(x)$.

Question 17

If $P(x) = x^4 + 4x - 2$ is divided by $x - 1$, what is the remainder?

Accepted Answer

$P(1) = 1^4 + 4(1) - 2 = 1 + 4 - 2 = 3$

Question 18

A business uses $B(x) = x^2 - 7x + 12$ to model its revenue. If $x = 3$ is a root, which of the following is also a root?

Accepted Answer

$x = 4$

Polynomial Division, Remainder Theorem, and Synthetic Division

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