Q: Given the polynomial $$p(x) = -x^2$ (x^2$ + 2)(x + 4)$$, which statement best describes its end behavior?

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

Question 1

Given the polynomial $$p(x) = -x^2$ (x^2$ + 2)(x + 4)$$, which statement best describes its end behavior?

Accepted Answer

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

Question 2

For the function $$g(x) = -\frac{1}{5}(x+5)(x-4)^2$, what is the multiplicity of the zero at x = -5$?

Accepted Answer

Multiplicity 1

Question 3

Which of the following is a correct application of the Rational Zeros Theorem for $$p(x) = 3x^4$ + 5x^3$ + 25x^2$ + 45x - 18$$?

Accepted Answer

Possible rational zeros are $$\pm1, \pm2, \pm3, \pm6, \pm9, \pm18, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{3}{3}, \pm\frac{6}{3}, \pm\frac{9}{3}, \pm\frac{18}{3}$$

Question 4

Using Descartes' Rule of Signs, how many positive real zeros does $$p(x) = 3x^4$ + 5x^3$ + 25x^2$ + 45x - 18$$ possibly have?

Accepted Answer

3 or 1

Question 5

Given $$p(x) = x^5$ - 3x^2$ + 4x + 2$$, use the Intermediate Value Theorem to show there is a zero between $$x = -1$$ and $$x = 0. $$Which calculation supports this?

Accepted Answer

$$p(-1) \u003e 0$$ and $$p(0) \u003c 0$$

Question 6

Write a degree 5 polynomial with real coefficients that has zeros at $$4 ($$multiplicity 2), $$-2$$, and $$5i. $$Which of the following is a valid form (with $$i$$ allowed)?

Accepted Answer

$$p(x) = (x-4)^2$(x+2)(x-5i)(x+5i)$$

Question 7

For the rational function $$r(x) = \frac{3x^2$ + 12x}{x^2$ - x - 6}$$, which of the following is a vertical asymptote?

Accepted Answer

$$x = 3$$

Question 8

What is the domain of $$f(x) = \log(4 - x^2$)$$?

Accepted Answer

$$-2 \u003c x \u003c 2$$

Question 9

For $$f(x) = \ln(x^2$)$$, which of the following best describes its domain?

Accepted Answer

$$x 
eq 0$$

Question 10

Given $$f(x) = -\log(9-x) - 2$$, what is the vertical asymptote?

Accepted Answer

$$x = 9$$

Question 11

For the rational function $$r(x) = \frac{3x^2$ + 12x}{x^2$ - x - 6}$$, which $$x-$$value creates a hole in the graph?

Accepted Answer

$$x = -2$$

Question 12

Which of the following describes the behavior of $$h(x) = \frac{1}{2}(x-3)(x+5)^3$ at x = -5$?

Accepted Answer

The graph touches and flattens at $$x = -5$$

Question 13

If $$p(x) = 3x^4$ + 5x^3$ + 25x^2$ + 45x - 18$$ has a complex zero at $$x = 2 + i$$, what other zero must it have?

Accepted Answer

$$x = 2 - i$$

Question 14

Which of the following is the $$y-$$intercept of $$r(x) = \frac{3x^2$ + 12x}{x^2$ - x - 6}$$?

Accepted Answer

$$y = 0$$

Question 15

For $$f(x) = \log(x^2$)$$, what is the value of $$f(-1)$$?

Accepted Answer

$$0$$

Precalculus Exam Review: Polynomial, Rational, and Logarithmic Functions

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