Q: Which of the following is the correct end behavior for the polynomial $$f(x) = -2x^4$ + 3x^2$ - x + 7$$?

$$y \to -\infty as$$ $$x \to \infty$$, $$y \to -\infty as$$ $$x \to -\infty$$

Question 1

Solve the quadratic equation $$x^2 + 6x + 41 = 0$ using the quadratic formula.

Accepted Answer

$$x = -3 \pm 4i$$

Question 2

Which of the following best describes the solutions to the equation $$x^2 + 6x + 41 = 0$?

Accepted Answer

Two different imaginary-number solutions

Question 3

Given the function $$f(x) = x^2$ - 2x + 1$$, find the vertex.

Accepted Answer

$$(1, 0)$$

Question 4

For the quadratic function $$f(x) = -x^2$ + 4x + 12$$, determine whether the function has a maximum or minimum value and find that value.

Accepted Answer

Maximum at $$x = 2$$, value $$16$$

Question 5

Find the intervals on which the function $$f(x) = x^2$ - 2x + 1 is $$increasing.

Accepted Answer

$$(1, \infty)$$

Question 6

A floating island is represented by the function $$P(x) = x^2$ + 5x - 24$$, where $$P(x) is $$the profit and $$x is $$the number of bungalows built. How many bungalows must be built to maximize profit?

Accepted Answer

$$x = 2.5$$

Question 7

Find the zeros of the polynomial $$f(x) = x^3$ - 2x^2$ - x + 2$$ and state the multiplicity of each.

Accepted Answer

$$x = -1 ($$multiplicity 1), $$x = 1 ($$multiplicity 1), $$x = 2 ($$multiplicity 1)$

Question 8

For the function $$f(x) = x^4$ - 2x^2$ + 1$$, what is the maximum number of real zeros the function can have?

Accepted Answer

4

Question 9

State the domain of the rational function $$f(x) = \frac{2x}{x^2$ - 9}.$$

Accepted Answer

$$x 
eq -3, x 
eq 3$$

Question 10

Find the vertical asymptote(s) of the rational function $$f(x) = \frac{2x}{x^2$ - 9}.$$

Accepted Answer

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

Question 11

Find the horizontal asymptote, if any, of the rational function $$f(x) = \frac{2x}{x^2$ - 9}.$$

Accepted Answer

$$y = 0$$

Question 12

Solve the inequality $$x^3$$ + $$2x^2 - 3x$$ \u003e 0$$.

Accepted Answer

$$(-3, 0) \cup (1, \infty)$$

Question 13

Given $$f(x) = x^3$ - 2x^2$ - x + 2$$, use synthetic division to determine whether $$x - 1 is a $$factor of $$f(x).$$

Accepted Answer

Yes, because the remainder is 0

Question 14

For the rational function $$f(x) = \frac{x^2$ - 1}{x - 1}$$, what is the equation of the slant (oblique) asymptote?

Accepted Answer

$$y = x + 1$$

Question 15

Evaluate $$f(x) = x^3$ - 2x^2$ - x + 2 at$$ $$x = 1$$ and $$x = 2$$, and use the Intermediate Value Theorem to determine if there is a root between $$x = 1$$ and $$x = 2.$$

Accepted Answer

Yes, because $$f(1)$$ and $$f(2)$$ have opposite signs

Question 16

Solve the inequality $$\frac{x^2$ - 3x + 2}{x - 1} \u003e 0.$$

Accepted Answer

$$(-\infty, 1) \cup (2, \infty)$$

Question 17

Given the polynomial $$f(x) = x^4$ - 2x^2$ + 1$$, how many turning points can its graph have at most?

Accepted Answer

3

Question 18

Which of the following is the correct end behavior for the polynomial $$f(x) = -2x^4$ + 3x^2$ - x + 7$$?

Accepted Answer

$$y 	o -\infty as$$ $$x 	o \infty$$, $$y 	o -\infty as$$ $$x 	o -\infty$$

College Algebra Exam 2 Review: Quadratic and Polynomial Functions, Rational Functions, and Inequalities

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