As $$x \to -\infty$$, $$f(x) \to +\infty$$

$$q(a) = 0$$ and $$p(a) \neq 0$$

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

Question 1

Which of the following statements is TRUE about the graph of any polynomial function $$f(x)$$?

Accepted Answer

It is continuous and smooth for all real $$x.$$

Question 2

Given the polynomial $$f(x) = x^4$ - 5x^2$ + 4$$, what is the end behavior of its graph as $$x 	o -\infty$$?

Accepted Answer

As $$x 	o -\infty$$, $$f(x) 	o +\infty$$

Question 3

Suppose $$f(x) = (x-2)^3$(x+1)^2. $$Which of the following is TRUE about the zeros of $$f(x)$?

Accepted Answer

$$x = 2 is a $$zero of multiplicity 3 and $$x = -1 is a $$zero of multiplicity 2

Question 4

A power function $$f(x) = x^n is $$symmetric with respect to the y-axis if:

Accepted Answer

$$n is $$even

Question 5

Which of the following best describes the number of turning points for a polynomial of degree $$n$$?

Accepted Answer

At most $$n-1$$ turning points

Question 6

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

Accepted Answer

There is a hole at $$x = 2$$

Question 7

A rational function $$R(x) = \frac{p(x)}{q(x)}$$ has a vertical asymptote at $$x = a if$$:

Accepted Answer

$$q(a) = 0$$ and $$p(a) 
eq 0$$

Question 8

Consider the function $$f(x) = -2x^6$ + 2x^2$. Which of the following describes its end behavior as x$$ 	o \infty$$?

Accepted Answer

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

Question 9

If $$f(x) = x^3$ - 4x^2$ + x - 6$$, what is the smallest possible degree of $$f(x)$$?

Accepted Answer

3

Question 10

Which of the following is the correct definition of a zero of a polynomial $$f(x)$$?

Accepted Answer

A value $$c$$ such that $$(x-c) is a $$factor of $$f(x)$$

Question 11

Given the rational function $$R(x) = \frac{x^2$+1}{x^2$-4}$$, what are the vertical asymptotes?

Accepted Answer

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

Question 12

Suppose you fit a quadratic polynomial to model the data for coal consumption over time. What is the general form of the quadratic polynomial?

Accepted Answer

$$f(x) = ax^2$ + bx + c$$

Question 13

If a rational function $$R(x)$$ has a hole at $$x = a$$, which of the following must be true?

Accepted Answer

$$p(a) = 0$$ and $$q(a) = 0$$

Question 14

Which of the following is TRUE about the horizontal asymptote of $$R(x) = \frac{2x^2$ + 1}{x^2$ - 3}$$?

Accepted Answer

The horizontal asymptote is $$y = 2$$

Question 15

A polynomial of degree $$n$$ has how many complex zeros, counting multiplicity?

Accepted Answer

Exactly $$n$$ complex zeros

Question 16

Given $$f(x) = x^5$, which of the following describes its symmetry?

Accepted Answer

Symmetric with respect to the origin

Question 17

Suppose $$f(x) = x^2$ + 4x + 3. $$What are the real zeros of $$f(x)$$?

Accepted Answer

$$x = -3$$ and $$x = 1$$

Question 18

Which of the following is NOT a feature of a polynomial function?

Accepted Answer

The graph has sharp corners

Question 19

If $$f(x) = x^3$ - 3x^2$ + 2x$$, what is the sum of the multiplicities of all its real zeros?

Accepted Answer

3

Polynomial and Rational Functions: Concepts, Properties, and Applications

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