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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