Question 1

Use polynomial long division to rewrite the following fraction in the form $$q(x) + \frac{r(x)}{d(x)}$$:

$$\frac{x^3$ - 5x^2$ + 5x - 3}{x^2$ - x + 1}$$

Accepted Answer

$$x - 5 + \frac{2x + 2}{x^2$ - x + 1}$$

Question 2

Given $$f(x) = x^3$ + 4x^2$ + x - 6$$ and that $$-2 is a $$zero, which of the following lists all remaining zeros?

Accepted Answer

$$-3, 1$$

Question 3

Find the vertical, horizontal, and slant asymptotes of $$f(x) = \frac{x^2$ - 4}{x - 3}. $$Which is correct?

Accepted Answer

Vertical: $$x = 3$$, Horizontal: none, Slant: $$y = x + 3$$

Question 4

Solve the inequality $$x^2$$ \geq 13x - 36$$ and express your answer in interval notation.

Accepted Answer

$$(-\infty, 4] \cup [9, \infty)$$

Question 5

Evaluate $$\log_2$ 32.$$

Accepted Answer

$$5$$

Question 6

Expand and simplify $$\log_7$ 49x^6$ using properties of logarithms.

Accepted Answer

$$2 + 6\log_7$ x$$

Question 7

Rewrite as a single logarithm: $$\ln 5 + 3\ln x - 8\ln y$$

Accepted Answer

$$\ln \left( \frac{5x^3$}{y^8$} \right)$$

Question 8

Solve for $$x$$: $$5^x \cdot 2^x = 25$$

Accepted Answer

$$x = 1$$

Question 9

Solve the logarithmic equation: $$\log_4$ (2x - 5) = 3$$

Accepted Answer

$$x = 16$$

Question 10

Find the exact solution for $$e^{4x} = 11$.

Accepted Answer

$$x = \frac{\ln 11}{4}$$

Question 11

Which of the following is the correct interval solution to $$x(x + 6)(x - 2) \u003c 0$$?

Accepted Answer

$$(-\infty, -6) \cup (0, 2)$$

Question 12

Given $$g(x) = \frac{x^3$}{x^2$ - x + 2}$$, what are the equations of the vertical and horizontal asymptotes?

Accepted Answer

Vertical: $$x = 2$$, Horizontal: $$y = 0$$

Question 13

Solve for $$x$$: $$\log_3$ (x + 2) = \log_3$ (8x - 5)$$

Accepted Answer

$$x = 1$$

Question 14

Expand and simplify $$\ln \left( \frac{11x^3$}{y^7$} \right).$$

Accepted Answer

$$\ln 11 + 3\ln x - 7\ln y$$

Question 15

Solve for $$x$$: $$2e^{5x} = 24$

Accepted Answer

$$x = \frac{\ln 12}{5}$$

Question 16

Rewrite as a single logarithm: $$\log_2$ (x^2$ + 9x + 20) - \log_2$ (x + 5)$$

Accepted Answer

$$\log_2$ (x + 4)$$

Question 17

Solve for $$x$$: $$3^x = 16$$

Accepted Answer

$$x = \frac{\ln 16}{\ln 3}$$

Question 18

Which of the following is the correct expansion for $$\ln \left( \frac{5x^6$}{y^5$} \right)$$?

Accepted Answer

$$\ln 5 + 6\ln x - 5\ln y$$

Question 19

Solve for $$x$$: $$\log_2$ (x^2$ - 36) = \log_2$ (5x)$$

Accepted Answer

$$x = 9$$

Precalculus Test 3 Review – Step-by-Step Study Guidance

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