$$f(x) \to -\infty as$$ $$x \to \pm \infty$$

Question 1

Which of the following is a possible rational zero of the polynomial $$f(x) = x^3$ - x^2$ - 37x + 35$$ according to the Rational Root Theorem?

Accepted Answer

$$\pm 1, \pm 35, \pm 5, \pm 7$$

Question 2

Given $$f(x) = x^2$ - 4x + 3$$, which of the following is a zero of $$f(x)$$?

Accepted Answer

$$x = 1$$

Question 3

If $$f(x) = x^3$ - 3x^2$ + 3x - 1$$, what is the end behavior of the graph as $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o \infty$$

Question 4

Which of the following polynomials has a double root at $$x = -4$$?

Accepted Answer

$$(x+4)^2$$

Question 5

Given $$f(x) = x^3$ + 2x^2$ - 5x - 6$$, use synthetic division to determine if $$x = -3 is a $$zero of $$f(x). $$What is the remainder?

Accepted Answer

$$0$$

Question 6

For the polynomial $$f(x) = x^4$ - 8x^2$ + 16$$, which of the following describes the number and type of real zeros?

Accepted Answer

Two real zeros, each with multiplicity 2

Question 7

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, which of the following is NOT a zero of $$f(x)$$?

Accepted Answer

$$x = 4$$

Question 8

Given $$f(x) = x^2$ + 4x + 5$$, what is the vertex of the parabola?

Accepted Answer

$$( -2, 1 )$$

Question 9

Which of the following describes the end behavior of $$f(x) = -2x^4$ + 5x^2$ - 1$$?

Accepted Answer

$$f(x) 	o -\infty as$$ $$x 	o \pm \infty$$

Question 10

Given the rational function $$f(x) = \frac{x^2$ - 9}{x^2$ - x - 6}$$, which of the following is a vertical asymptote?

Accepted Answer

$$x = 2$$

Question 11

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

Accepted Answer

$$x + 1$$

Question 12

If $$f(x) = x^2$ - 6x + 8$$, what is the sum of the zeros of $$f(x)$$?

Accepted Answer

$$6$$

Question 13

Given $$f(x) = x^2$ - 2x + 5$$, what is the discriminant and what does it tell you about the zeros?

Accepted Answer

Discriminant is $$-16$$, so there are two complex zeros

Question 14

Which of the following is the correct vertex form for $$f(x) = x^2$ - 6x + 18$$?

Accepted Answer

$$f(x) = (x-3)^2$ + 9$$

Question 15

For the rational function $$f(x) = \frac{(x+1)(3x-2)}{(x-3)(x+3)}$$, what is the domain?

Accepted Answer

$$(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$$

Question 16

If $$f(x) = x^3$ - 4x^2$ + 5x - 2$$, use the Intermediate Value Theorem to determine if there is a zero between $$x = 0$$ and $$x = 1.$$

Accepted Answer

Yes, because $$f(0)$$ and $$f(1)$$ have opposite signs

Question 17

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

Accepted Answer

$$x = 1$$

Question 18

For the polynomial $$f(x) = x^3$ - 2x^2$ - x + 2$$, what is the sum of the coefficients?

Accepted Answer

$$0$$

Question 19

Which of the following describes the slant asymptote of $$f(x) = \frac{x^2$ + 5x + 10}{x + 3}$$?

Accepted Answer

$$y = x + 2$$

Question 20

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, which of the following is the factorization of $$f(x)$$?

Accepted Answer

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

Polynomial and Rational Functions: Zeros, Factorization, and Graph Behavior

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