Question 1

According to the Factor Theorem, which of the following statements is true for a polynomial function $$f(x)$$?

Accepted Answer

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

Question 2

Given $$f(x) = 6x^3$ + 19x^2$ + 2x - 3$$, which of the following is a factor of $$f(x)$$?

Accepted Answer

$$x+3$$

Question 3

If $$f(x) = x^3$ - 3x^2$ - 5x + 6$$ and $$x=1 is a $$zero, which of the following is the correct factorization of $$f(x)$$?

Accepted Answer

$$(x-1)(x^2-3x-6$$)$$

Question 4

Which theorem states that if $$a+bi is a $$zero of a polynomial with real coefficients, then $$a-bi is $$also a zero?

Accepted Answer

Conjugate Zeros Theorem

Question 5

For the polynomial $$p(x) = x^4$ + 16x^2$ - 225$$, given that $$5i is a $$zero, which of the following is also a zero?

Accepted Answer

$$-5i$$

Question 6

Using the Rational Root Theorem, what are the possible rational zeros of $$f(x) = 6x^3$ + 7x^2$ - 12x^2$ - 3x + 2$$?

Accepted Answer

$$\pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{2}{3}$$

Question 7

Which of the following is the correct factorization of $$f(x) = x^3$ - 5x^2$ + 2x + 8$$ into linear factors?

Accepted Answer

$$(x-2)(x-4)(x+1)$$

Question 8

If $$f(x) = x^3$ + 2x^2$ - 3x - 6$$ and $$x=-3 is a $$zero, which of the following is NOT a correct statement?

Accepted Answer

$$x-3 is a $$factor of $$f(x)$$

Question 9

A polynomial $$f(x)$$ has real coefficients and $$2+5i is a $$zero. Which of the following must also be a zero?

Accepted Answer

$$2-5i$$

Question 10

Which of the following is a correct application of the Factor Theorem?

Accepted Answer

If $$f(2) = 0$$, then $$x-2 is a $$factor of $$f(x)$$

Question 11

Given $$f(x) = x^3$ - 3x^2$ - 5x + 6$$, which of the following is NOT a possible rational zero according to the Rational Root Theorem?

Accepted Answer

$$4$$

Question 12

If $$f(x) = x^3$ + 2x^2$ - 3x - 6$$ and you perform synthetic division by $$x+3$$, what is the remainder?

Accepted Answer

$$0$$

Question 13

Which of the following best describes the process of factoring a cubic polynomial using the Factor Theorem?

Accepted Answer

Find a zero $$k$$ such that $$f(k)=0$$, then divide $$f(x) by$$ $$x-k to $$get a quadratic factor

Question 14

If $$f(x) = x^3$ - 5x^2$ + 2x + 8$$ and $$x=2 is a $$zero, what is the next step to factor $$f(x)$$ completely?

Accepted Answer

Divide $$f(x) by$$ $$x-2 to $$get a quadratic factor

Question 15

Which of the following is the correct set of possible rational zeros for $$f(x) = 6x^3$ + 7x^2$ - 12x^2$ - 3x + 2$$ using the Rational Root Theorem?

Accepted Answer

$$\pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{2}{3}$$

Polynomial Functions: Factor Theorem, Rational Zeros, and Conjugate Zeros Theorem

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