Question 1

Which of the following statements best describes the Factor Theorem for a polynomial function $$f(x)$$?

Accepted Answer

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

Question 2

Given $$f(x) = x^2$ - 6x - 7$$, which of the following is NOT a zero of $$f(x)$$?

Accepted Answer

$$-7$$

Question 3

Suppose $$f(x) = 6x^3$ + 19x^2$ + 2x - 3$$ and $$-3 is a $$zero. Which of the following is a correct linear factor of $$f(x)$$?

Accepted Answer

$$(x + 3)$$

Question 4

The Rational Zeros Theorem states that if $$\frac{p}{q} is a $$rational zero of a polynomial with integer coefficients, then:

Accepted Answer

$$p is a $$factor of the constant term and $$q is a $$factor of the leading coefficient.

Question 5

Given $$f(x) = 6x^4$ + 7x^3$ - 12x^2$ - 3x + 2$$, what is the complete list of possible rational zeros according to the Rational Zeros Theorem?

Accepted Answer

$$\pm1, \pm2, \pm\frac{1}{2}, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{1}{6}, \pm\frac{2}{6}$$

Question 6

If a polynomial $$f(x) of $$degree 4 has zeros at $$x = 2 ($$multiplicity 2), $$x = -1$$, and $$x = 3$$, which of the following is the correct factored form?

Accepted Answer

$$(x - 2)^2$(x + 1)(x - 3)$$

Question 7

According to the Fundamental Theorem of Algebra, how many zeros (real or complex) does a degree $$n$$ polynomial have?

Accepted Answer

Exactly $$n$$ zeros, counting multiplicities.

Question 8

A polynomial $$P(x) of $$degree 5 has a zero at $$x = 1$$ with multiplicity 3. Which statement is true?

Accepted Answer

$$x = 1 is a $$zero of $$P(x)$$, and $$(x - 1)^3$ is a factor.

Question 9

If $$f(x) = ax^3$ + bx^2$ + cx + d$$ and $$f(2) = 0$$, which of the following must be true?

Accepted Answer

$$(x - 2) is a $$factor of $$f(x).$$

Question 10

A cubic polynomial $$f(x)$$ has zeros at $$x = -1, 2, 4$$ and $$f(1) = 3. $$What is the value of the leading coefficient $$a in$$ $$f(x) = a(x + 1)(x - 2)(x - 4)$$?

Accepted Answer

$$a = \frac{3}{15}$$

Question 11

Which of the following best describes the multiplicity of a zero?

Accepted Answer

The number of times a specific zero appears in the completely factored form.

Question 12

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

Accepted Answer

$$x = 1, -1, 2, -2$$

Question 13

A chemist models the concentration of a substance with $$C(t) = t^3$ - 4t^2$ + 4t. At $$what time $$t$$ does the concentration reach zero?

Accepted Answer

$$t = 0, 2$$

Question 14

If $$f(x) = 2x^3$ - 3x^2$ - 8x + 12$$, which of the following is a possible rational zero?

Accepted Answer

$$\frac{3}{2}$$ and $$\frac{6}{1}$$

Question 15

A polynomial $$f(x) of $$degree 3 has only one real zero at $$x = -2$$ with multiplicity 3. Which of the following is $$f(x) up to a $$constant multiple?

Accepted Answer

$$(x + 2)^3$

Question 16

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

Accepted Answer

$$(x + 2)$$

Question 17

A medical researcher uses $$P(x) = x^2$ - 4x + 3 to $$model the probability of a reaction. For which value(s) of $$x is $$the probability zero?

Accepted Answer

$$x = 1, 3$$

Question 18

If $$f(x) = x^4$ + 2x^3$ - 7x^2$ - 8x + 12$$, what is the maximum number of distinct real zeros $$f(x)$$ can have?

Accepted Answer

4

Question 19

Which of the following is the correct complete factorization of $$f(x) = x^2$ - 9$$ over the real numbers?

Accepted Answer

$$(x - 3)(x + 3)$$

Zeros of Polynomial Functions (Section 3.3) – Study Notes

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