Question 1

Factor the following polynomial: $$14p^2 - 19pq$$ - $$3q^2$$

Accepted Answer

$$(7p + q)(2p - 3q)$$

Question 2

Simplify the following expression and evaluate using exact values: $$(\sqrt{23} + \sqrt{29})^2$

Accepted Answer

$$26 + 2\sqrt{667}$$

Question 3

Perform multiplication in the given expression: $$b^2(5b - 3)(7b + 8$$)$$

Accepted Answer

$$35b^4$$ - $$19b^3$$ - $$24b^2$$

Question 4

Simplify the following expression and write the answer in standard form: $$(i^{26}$$ - $$i^{122})/i$$^{63}$

Accepted Answer

$$2$$

Question 5

Solve by factoring: $$7a(a - 4) = 4a^2$ - 21a$$

Accepted Answer

$$a = 0, 3/7$$

Question 6

By using the quadratic formula, solve the following equation: $$x^2 - 16x + 73 = 0$

Accepted Answer

$$x = 8 - 3i, 8 + 3i$$

Question 7

Enlist all values of $$x$$ that will make the given expression undefined: $$\frac{5}{x^2$ - 7x + 6}$$

Accepted Answer

$$\{1, 6\}$$

Question 8

Identify if the given statement is correct or not. If not, make necessary amends on the right side of the equation to make it correct: $$(17 + 3i) - (6 + 13i) = 11 + 16i$$

Accepted Answer

Incorrect statement: Correct statement should be $$(17 + 3i) - (6 + 13i) = 11 - 10i$$

Question 9

Find the domain of the following square root function: $$\sqrt{(10x - 5) - 3}$$

Accepted Answer

$$[1/2, 15/29)$$

Question 10

In the given equation, complete the square and solve it: $$18x^2 + x = 4$

Accepted Answer

Solution set: $$\{-1/2, 4/9\}$$

Question 11

For the function $$y = (x - 3)^2$ - 9$$, identify its x-intercepts and match it with the correct graph.

Accepted Answer

x-intercepts $$0, 6$$

Question 12

Using the given relationship, write an expression and solve the equation for $$N$$. $$M$$ varies directly as $$O$$ and inversely as the cube root of $$N.$$

Accepted Answer

$$N = kO^3$/M^3$

Question 13

Graph the following equation and label the x-intercepts, if there are any: $$y = 4(x + 2)^2$ + 7(x + 2) - 2$$

Accepted Answer

x-intercepts $$(-2.25, 0)$$ and $$(0, 0)$$

Question 14

For the given conditions: Passing through $$(-2, 0)$$ and $$(4, -9)$$, write an equation for the line in slope-intercept form by first going through the point-slope form.

Accepted Answer

$$y = -3x/2 - 3$$

Question 15

Use the standard graph of the given function $$f(x) = |x| to $$transform it into the following function $$g(x) = 2|4x - 3| + 3. $$Which graph matches the transformation?

Accepted Answer

Vertex at $$(0.75, 3)$$, steeper slope

Question 16

Given the two functions $$f(x) = 71x - 2$$, $$g(x) = x + 10$$, find $$fg$$ and identify the domain.

Accepted Answer

$$fg(x) = 71x^2$ + 712x - 20$$; Domain: $$(-\infty, \infty)$$

Question 17

For the given one-to-one function, find the inverse function, $$f^{-1}(x$$)$$, and verify that your equation is correct. f(x) = x/2$

Accepted Answer

$$f^{-1}(x) = 2x$

Question 18

Solve the following equation by completing the square: $$y^2 - 8y - 3 = 0$

Accepted Answer

$$y = 4 \pm \sqrt{19}$$

Question 19

Determine a quadratic function $$f(x)$$ for the graph given below with vertex at $$(3, 9)$$ and opening upwards.

Accepted Answer

$$f(x) = x^2$ - 6x + 9$$

Question 20

Justify that the polynomial $$f(x) = x^3$ + 4x - 3$$ has a real zero in between $$0$$ and $$1 by $$using the Intermediate Value Theorem.

Accepted Answer

Since $$f(0) = -3$$ and $$f(1) = 2$$, the function has a real zero in between $$0$$ and $$1.$$

Precalculus Study Guide: Algebraic Expressions, Functions, and Graphs

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