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

Both ends go up as $$x \to \pm \infty$$

Question 1

Which of the following is the correct definition of a polynomial function of degree $$n$$?

Accepted Answer

$$f(x) = a_n x^n + a_{n-1} x$$^{n-1}$$ + \ldots + a_1$ x + a_0$, where a_n$$ 
eq 0$$

Question 2

Given the polynomial $$f(x) = (x-3)^2$(x+4)$$, what is the multiplicity of the zero at $$x=3$$?

Accepted Answer

2

Question 3

Which of the following statements about the end behavior of $$f(x) = -2x^5$ + 3x^2$ - 7 is $$correct?

Accepted Answer

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

Question 4

If $$f(x) = x^4$ - 5x^2$ + 6$$, how many real zeros can $$f(x)$$ have at most?

Accepted Answer

4

Question 5

Which theorem states that a polynomial function of degree $$n$$ has at most $$n$$ turning points?

Accepted Answer

Turning Points Theorem

Question 6

Given $$f(x) = x^3$ - 4x^2$ + x - 6$$, use the Factor Theorem to determine if $$x=2 is a $$zero of $$f(x).$$

Accepted Answer

Yes, because $$f(2) = 0$$

Question 7

Which of the following is a possible rational zero of $$f(x) = 2x^3$ - x^2$ - x + 4$$ according to the Rational Zeros Theorem?

Accepted Answer

$$\pm 1, \pm 2, \pm 4, \pm \frac{1}{2}, \pm \frac{2}{2}, \pm \frac{4}{2}$$

Question 8

For the rational function $$R(x) = \frac{2x^2$ - 4}{x}$$, what is the domain of $$R(x)$$?

Accepted Answer

All real numbers except $$x=0$$

Question 9

Which of the following describes the horizontal asymptote of $$R(x) = \frac{3x^2$ - 1}{x^2$ - 9}$$?

Accepted Answer

$$y=3$$

Question 10

A polynomial $$f(x) of $$degree 5 with real coefficients has zeros $$2, 4, 3+4i. $$What are the remaining zeros?

Accepted Answer

$$3-4i$$ and 0$

Question 11

Which property must the graph of a polynomial function always have?

Accepted Answer

It is always smooth and continuous

Question 12

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

Accepted Answer

5

Question 13

If $$R(x) = \frac{x^2$ - 4}{x}$$, what is the vertical asymptote of $$R(x)$$?

Accepted Answer

$$x=0$$

Question 14

Which theorem guarantees that a polynomial function of degree $$n \geq 1$$ has at least one complex zero?

Accepted Answer

Fundamental Theorem of Algebra

Question 15

For the rational function $$R(x) = \frac{2x^3$ + 5x^2$ - 4}{x^2$ + x - 6}$$, what is the degree of the numerator?

Accepted Answer

3

Question 16

If $$f(x) = x^4$ - 2x^2$ + 1$$, which of the following describes the end behavior of $$f(x)$$?

Accepted Answer

Both ends go up as $$x 	o \pm \infty$$

Question 17

Which of the following is true about the number of real zeros of a polynomial function?

Accepted Answer

It can have at most as many real zeros as its degree

Question 18

If $$f(x) = x^3$ - 2x^2$ + x - 2 is $$divided by $$x-1$$, what is the remainder?

Accepted Answer

-3

Question 19

Which of the following is the correct application of 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 $$b$$

Polynomial and Rational Functions: Definitions, Properties, and Theorems

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