As $$x \to \infty$$, $$f(x) \to -\infty$$; as $$x \to -\infty$$, $$f(x) \to -\infty$$

Question 1

Which of the following best describes the end behavior of the polynomial function $$f(x) = -2x^4$ + 5x^2$ - 3$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o -\infty$$; as $$x 	o -\infty$$, $$f(x) 	o -\infty$$

Question 2

Given the rational function $$R(x) = \frac{x^2$ - 4}{x^2$ - 1}$$, which of the following statements is true?

Accepted Answer

The function has a horizontal asymptote at $$y = 0$$

Question 3

Which of the following is the correct domain of the rational function $$f(x) = \frac{2x+1}{x^2$-9}$$?

Accepted Answer

$$x 
eq 3,\ x 
eq -3$$

Question 4

A company models its profit with the polynomial $$P(x) = x^3$ - 6x^2$ + 11x - 6. $$What are the real zeros of this function?

Accepted Answer

$$x = 1,\ x = 2,\ x = 3$$

Question 5

Which of the following describes the vertical asymptotes of $$f(x) = \frac{x+2}{x^2$-5x+6}$$?

Accepted Answer

$$x = 2,\ x = 3$$

Question 6

Solve the inequality $$(x-4)(x+1) \geq 0$$ and select the correct solution set.

Accepted Answer

$$x \leq -1 or$$ $$x \geq 4$$

Question 7

Which of the following is a correct application of the Rational Root Theorem for $$f(x) = 2x^3$ - 3x^2$ - 11x + 6$$?

Accepted Answer

Possible rational roots are $$\pm1,\ \pm2,\ \pm3,\ \pm6,\ \pm\frac{1}{2},\ \pm\frac{3}{2}$$

Question 8

If $$f(x) = x^4$ + 2x^2$ + 1$$, what is the minimum value of $$f(x)$$ for all real $$x$$?

Accepted Answer

$$1 at$$ $$x = 0$$

Question 9

Which of the following is the correct horizontal asymptote for $$f(x) = \frac{3x^2$ + 2}{2x^2$ - 5}$$?

Accepted Answer

$$y = \frac{3}{2}$$

Question 10

Solve for all complex solutions of $$x^2 + 4 = 0$.

Accepted Answer

$$x = 2i,\ x = -2i$$

Question 11

A rational function $$R(x)$$ has a numerator of degree 2 and a denominator of degree 3. What is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 12

Which of the following is a true statement about the Fundamental Theorem of Algebra?

Accepted Answer

Every polynomial equation of degree $$n$$ has exactly $$n$$ complex roots, counting multiplicities.

Question 13

Given $$f(x) = x^3$ - 2x^2$ - x + 2$$, use synthetic division to test if $$x = 2 is a $$root.

Accepted Answer

The remainder is $$0$$, so $$x = 2 is a $$root.

Question 14

Which of the following best describes the graph of $$f(x) = \frac{x-1}{x+2}$$?

Accepted Answer

Vertical asymptote at $$x = -2$$, horizontal asymptote at $$y = 0$$

Question 15

Solve the inequality $$\frac{x-2}{x+1} \u003e 0.$$

Accepted Answer

$$x \u003e 2 or$$ $$x \u003c -1$$

Question 16

Which of the following is a possible rational zero of $$f(x) = 6x^3$ + 5x^2$ - 2x + 1$$ according to the Rational Root Theorem?

Accepted Answer

$$\frac{1}{6}$$

Question 17

If $$f(x) = x^2$ - 4x + 4$$, what is the multiplicity of the zero at $$x = 2$$?

Accepted Answer

2

Question 18

Which of the following best describes the slant (oblique) asymptote of $$f(x) = \frac{x^2$ + 2x + 1}{x + 1}$$?

Accepted Answer

$$y = x + 1$$

Question 19

A rational function $$R(x)$$ has a hole at $$x = 2. $$Which of the following could be its equation?

Accepted Answer

$$R(x) = \frac{(x-2)(x+3)}{(x-2)(x-5)}$$

Chapter 5 Study Guide: Polynomial & Rational Functions

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