Question 1

According to the Remainder Theorem, what is the remainder when the polynomial $f(x) = x^3 - 2x^2 + 5x - 7$ is divided by $x - 2$?

Accepted Answer

$1$

Question 2

Which of the following statements best describes the Factor Theorem?

Accepted Answer

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

Question 3

Given $f(x) = 2x^3 + 3x^2 - 7x - 6$, which of the following is a potential rational zero according to the Rational Zeros Theorem?

Accepted Answer

$\frac{1}{2}$

Question 4

A polynomial $f(x)$ of degree $n$ has at most how many real zeros?

Accepted Answer

$n$

Question 5

If $f(x) = x^3 - 4x^2 + 5x - 2$ and $f(1) = 0$, what does this tell you about $x = 1$?

Accepted Answer

$x = 1$ is a zero and $(x - 1)$ is a factor of $f(x)$.

Question 6

Which of the following is NOT a possible rational zero of $f(x) = 3x^3 - 5x^2 + 2x - 8$?

Accepted Answer

$\frac{5}{3}$

Question 7

A chemist is modeling the concentration of a substance with $f(x) = x^3 - 6x^2 + 11x - 6$. If the concentration must be zero, which of the following values of $x$ is a solution?

Accepted Answer

$x = 1$

Question 8

If $f(x) = x^4 - 5x^2 + 4$, which of the following is a real zero of $f(x)$?

Accepted Answer

$x = 2$

Question 9

A business models its profit with $P(x) = x^3 - 2x^2 - 5x + 6$. Using the Rational Zeros Theorem, which of the following is NOT a possible rational zero?

Accepted Answer

$6$

Question 10

Given $f(x) = x^3 + 2x^2 - x - 2$, use the Factor Theorem to determine which of the following is a factor of $f(x)$.

Accepted Answer

$(x + 2)$

Question 11

A polynomial $f(x)$ is divided by $(x - 3)$ and the remainder is $5$. What is the value of $f(3)$?

Accepted Answer

$5$

Question 12

If $f(x) = x^3 - 7x + 6$, which of the following is NOT a real zero of $f(x)$?

Accepted Answer

$x = -2$

Question 13

A medical researcher uses $f(x) = x^2 - 4x + 3$ to model the spread of a virus. At what value of $x$ does the model predict zero cases?

Accepted Answer

$x = 1$

Question 14

Which of the following is a correct application of the Rational Zeros Theorem for $f(x) = 4x^3 - 3x^2 + 2x - 1$?

Accepted Answer

Possible rational zeros are $\pm1, \pm\frac{1}{2}, \pm\frac{1}{4}$.

Question 15

If $f(x) = x^3 - 2x^2 - 5x + 6$ and you know $x = 3$ is a zero, which of the following is a correct factorization step?

Accepted Answer

$f(x) = (x - 3)(x^2 + x - 2)$

Question 16

A polynomial $f(x)$ has degree 5. What is the maximum number of real zeros it can have?

Accepted Answer

$5$

Question 17

If $f(x) = x^3 - 4x^2 + 4x$, which of the following is NOT a real zero?

Accepted Answer

$x = 4$

Question 18

A company models its revenue with $R(x) = x^3 - 6x^2 + 11x - 6$. If $x$ represents the number of products sold, what is the minimum number of products that results in zero revenue?

Accepted Answer

$x = 1$

Question 19

Given $f(x) = x^3 - 2x^2 + x - 2$, which of the following is a correct application of the Remainder Theorem for $x = 2$?

Accepted Answer

The remainder is $f(2) = 0$.

3.6 The Real Zeros of a Polynomial Function: Remainder and Factor Theorems, Rational Zeros

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