Question 1

Which of the following best describes the end behavior of the polynomial $$f(x) = -2x^5$ + 3x^3$ - x + 7 as$$ $$x 	o \infty$$?

Accepted Answer

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

Question 2

Given the polynomial $$P(x) = (x-2)^3$(x+1)^2$$, what is the multiplicity of the zero at $$x = -1$ and how does the graph behave at this zero?

Accepted Answer

Multiplicity 2; graph touches but does not cross the x-axis

Question 3

Which of the following is the correct application of the Factor Theorem?

Accepted Answer

If $$f(a) = 0$$, then $$(x - a) is a $$factor of $$f(x)$$

Question 4

A rational function $$f(x) = \frac{x^2$ - 4}{x^2$ - 1}$$ has which of the following features?

Accepted Answer

Vertical asymptotes at $$x = 1$$ and $$x = -1$$

Question 5

Given $$g(x) = x^4$ - 5x^2$ + 4$$, what is the maximum number of turning points the graph of $$g(x)$$ can have?

Accepted Answer

3

Question 6

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

Accepted Answer

All real numbers except $$x = 3$$ and $$x = -3$$

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

A polynomial $$P(x)$$ has zeros at $$x = 2 ($$multiplicity 2), $$x = -1 ($$multiplicity 3), and $$x = 4 ($$multiplicity 1). Which statement is true about the graph at $$x = -1$$?

Accepted Answer

The graph crosses the x-axis at $$x = -1$$

Question 9

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

Accepted Answer

$$y = 3$$

Question 10

Using the Rational Zero Theorem, which of the following is a possible rational zero of $$f(x) = 2x^3$ - 3x^2$ + x - 6$$?

Accepted Answer

$$\frac{3}{2} or$$ $$-\frac{3}{2}$$

Question 11

If $$f(x) = x^3$ - 4x^2$ + 5x - 2 is $$divided by $$x - 2$$ using synthetic division, what is the remainder?

Accepted Answer

1

Question 12

Which of the following statements about the Fundamental Theorem of Algebra is correct?

Accepted Answer

Every polynomial of degree $$n$$ has at least one zero in the set of complex numbers.

Question 13

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

Accepted Answer

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

Question 14

For the rational function $$f(x) = \frac{x^2$ - 1}{x - 1}$$, what is true about $$x = 1$$?

Accepted Answer

There is a hole at $$x = 1$$

Question 15

Which of the following best describes the graph of a polynomial with a positive leading coefficient and odd degree?

Accepted Answer

Falls to the left and rises to the right

Question 16

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

Accepted Answer

4 or 2

Question 17

Which of the following is the correct process to find the domain of a rational function $$f(x) = \frac{p(x)}{q(x)}$$?

Accepted Answer

Set $$q(x) = 0$$ and exclude those $$x$$ values from the domain

Question 18

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

Accepted Answer

$$(x+1)$$

Question 19

If a polynomial has a zero at $$x = 3$$ with multiplicity 4, what does the graph do at $$x = 3$$?

Accepted Answer

Touches the x-axis and turns around

Question 20

Which of the following is true about the number of turning points of a polynomial of degree $$n$$?

Accepted Answer

It can have at most $$n-1$$ turning points

Polynomial and Rational Functions: Key Concepts and Techniques

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