Q: Use the Factor Theorem to determine whether $$x - \frac{1}{2} is a $$factor of $$f(x) = 2x^4$ - x^3$ + 2x - 1.$$

No, because $$f\left(\frac{1}{2}\right) \neq 0$$

Question 1

Given the graph of a polynomial function with zeros at $$x = -4$$, $$x = -2$$, and $$x = 2$$, where the graph touches the x-axis at $$x = -4$$ and $$x = 2$$ and crosses at $$x = -2$$, which of the following is the correct equation for $$f(x)$$? (Assume the leading coefficient is negative and the y-intercept is negative.)

$$A.\ f(x) = \frac{1}{8}(x+4)^2$(x+2)(x-2)^2$$

$$B.\ f(x) = \frac{1}{8}(x+4)(x+2)(x-2)$$

$$C.\ f(x) = -\frac{1}{8}(x+4)(x+2)(x-2)$$

$$D.\ f(x) = -\frac{1}{8}(x+4)(x+2)^2$(x-2)$$

$$E.\ f(x) = \frac{1}{8}(x+4)(x+2)(x-2)^2$

F$$.\ f(x) = \frac{1}{8}$$(x+4)(x+2)^2(x-2$$)$

Accepted Answer

$$f(x) = -\frac{1}{8}(x+4)(x+2)^2$(x-2)$$

Question 2

For the polynomial $$f(x) = -8\left(x+\frac{1}{4}\$$right)^2$$(x+8)^3$, what are the real zeros and their multiplicities? At which zeros does the graph touch or cross the x-axis?

Accepted Answer

Zeros: $$x = -\frac{1}{4} ($$multiplicity 2, touches), $$x = -8 ($$multiplicity 3, crosses)

Question 3

Given the rational function $$H(x) = \frac{16x}{x^2$ - 11x + 28}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 4$$ and $$x = 7$$

Question 4

For the rational function $$H(x) = \frac{x^3$ - 64}{4 - 5x + x^2$}$$, what is the horizontal asymptote?

Accepted Answer

None

Question 5

Use the Remainder Theorem to find the remainder when $$f(x) = 4x^6$ - 64x^4$ + x^3$ - 20 is $$divided by $$x + 4.$$

Accepted Answer

$$-4,180$$

Question 6

Use the Factor Theorem to determine whether $$x - \frac{1}{2} is a $$factor of $$f(x) = 2x^4$ - x^3$ + 2x - 1.$$

Accepted Answer

No, because $$f\left(\frac{1}{2}ight) 
eq 0$$

Question 7

List all possible rational zeros of $$f(x) = 2x^9$ - x^7$ + 9x^6$ + 44$$ using the Rational Zeros Theorem.

Accepted Answer

$$\pm1, \pm2, \pm4, \pm11, \pm22, \pm44, \pm\frac{1}{2}, \pm\frac{11}{2}, \pm\frac{22}{2}, \pm\frac{44}{2}$$

Question 8

Solve the inequality $$x^3$$ - $$4x^2 - 45x$$ \u003c 0$$.

Accepted Answer

$$(-\infty, -5) \cup (0, 9)$$

Question 9

Solve the inequality $$\frac{6x-5}{x+2} \leq 5.$$

Accepted Answer

$$(-\infty, -2) \cup \left(-2, \frac{15}{11}\right]$$

Question 10

Given $$R(x) = \frac{x^2$}{x^2$ + x - 30}$$, what is the domain of $$R(x)$$?

Accepted Answer

$$(-\infty, -6) \cup (-6, 5) \cup (5, \infty)$$

Question 11

For $$R(x) = \frac{x^2$}{x^2$ + x - 30}$$, what are the vertical asymptotes?

Accepted Answer

$$x = -6$$ and $$x = 5$$

Question 12

For $$R(x) = \frac{x^2$}{x^2$ + x - 30}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 1$$

Question 13

Find all real zeros of $$f(x) = 4x^4$ + 5x^3$ + 13x^2$ + 15x + 3$$ and factor $$f$$ over the real numbers.

Accepted Answer

No real zeros; $$f(x)$$ cannot be factored over the real numbers.

Question 14

Find all real zeros of $$f(x) = 5x^3$ - x^2$ + 5x - 1$$ and factor $$f$$ over the real numbers.

Accepted Answer

Real zero: $$x = 1$$; $$f(x) = (x-1)(5x^2$ + 5)$$

Question 15

Find the complex zeros of $$f(x) = x^3$ - 6x^2$ + 61x - 106$$ and write $$f in $$factored form.

Accepted Answer

Zeros: $$x = 1, x = 3 + 8i, x = 3 - 8i$$; $$f(x) = (x-1)(x-(3+8i))(x-(3-8i))$$

Question 16

Find the domain of the function $$g(x) = \sqrt{\frac{x-4}{x+10}}.$$

Accepted Answer

$$(-10, 4]$$

Exam 3 Review: Polynomial and Rational Functions (College Algebra)

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