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

Question 1

Which of the following is NOT a property of the graph of a polynomial function?

Accepted Answer

The graph can have sharp corners or cusps.

Question 2

Given the polynomial $$f(x) = 2x^4$ - 3x^2$ + 5$$, what is the degree and leading coefficient of $$f(x)$$?

Accepted Answer

Degree: 4, Leading coefficient: 2

Question 3

Which of the following functions is NOT a polynomial function?

Accepted Answer

$$g(x) = \sqrt{x}$$

Question 4

If $$f(x) = (x-2)^2$(x+1)$$, what is the multiplicity of the zero at $$x=2$$ and how does the graph behave at this zero?

Accepted Answer

Multiplicity 2; the graph touches the x-axis at $$x=2.$$

Question 5

A polynomial function of degree $$n$$ can have at most how many turning points?

Accepted Answer

$$n-1$$

Question 6

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

Accepted Answer

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

Question 7

Which of the following rational functions has a vertical asymptote at $$x=2$$?

Accepted Answer

$$R(x) = \frac{x+1}{x-2}$$

Question 8

Given $$R(x) = \frac{2x^2$ + 3}{x^2$ - 4}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 9

For the rational function $$R(x) = \frac{3x^2$ + 1}{x^2$ - 9}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 3$$

Question 10

Which of the following best describes the behavior of the graph of $$R(x) = \frac{1}{(x-1)^2$}$$ near $$x=1$$?

Accepted Answer

The graph approaches $$+\infty on $$both sides of $$x=1.$$

Question 11

Solve the inequality $$x^3 - 4x$$ \u003e 0$$. Which intervals are part of the solution set?

Accepted Answer

$$(-\infty, -2) \cup (0, 2)$$

Question 12

If $$f(x) = x^3$ - 2x^2$ - x + 2$$, use the Rational Zeros Theorem to list all possible rational zeros.

Accepted Answer

$$\pm1, \pm2$$

Question 13

Which of the following statements is TRUE about the zeros of a polynomial function with real coefficients?

Accepted Answer

If $$a + bi is a $$zero, then $$a - bi$$ must also be a zero.

Question 14

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

Accepted Answer

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

Question 15

A rational function $$R(x)$$ has a numerator of degree 3 and a denominator of degree 2. What type of asymptote does it have?

Accepted Answer

Oblique (slant) asymptote

Question 16

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

Accepted Answer

$$x = 4$$

Question 17

Which of the following best describes the Intermediate Value Theorem?

Accepted Answer

If $$f(a)$$ and $$f(b)$$ have opposite signs, there is at least one real zero between $$a$$ and $$b.$$

Question 18

Which of the following is the correct application of the Remainder Theorem for $$f(x) = x^3$ + 2x^2$ - x + 1$$ divided by $$x-1$$?

Accepted Answer

The remainder is $$f(1) = 1^3$ + 2(1)^2$ - 1 + 1 = 3.$$

Polynomial and Rational Functions: Precalculus Study Guide

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