Question 1

Which of the following numbers from the set $$A = \{36, 0.5, 0, \frac{1}{3}, -\pi, 4, 13, -12, 0.666...\} is an $$irrational number?

Accepted Answer

$$-\pi$$

Question 2

Evaluate the following expression: $$\frac{20}{16} - \frac{25}{5} + \frac{2}{4}. $$What is the result?

Accepted Answer

$$-3.25$$

Question 3

Given $$P(x) = x^4$ + 2x^2$ - 3x + 1$$ and $$Q(x) = x^2$ - 3$$, what is $$P(x) + Q(x)$$?

Accepted Answer

$$x^4$$ + $$3x^2 - 3x - 2$

Question 4

Using special products, what is the expanded form of $$(x + 2)^2$?

Accepted Answer

$$x^2 + 4x + 4$

Question 5

Factor the trinomial $$x^2 + 12x - 16$ completely over the integers.

Accepted Answer

Question 6

Which of the following trinomials from $$x^2 + 12x - 16$, x^2$ - 4x + 16$$, $$x^2 + 11x - 10$ will have a different factored form over the real numbers compared to factoring over the integers?

Accepted Answer

$$x^2 - 4x + 16$

Question 7

Factor completely over the real numbers: $$x^3$$ - $$64y^3.$$

Accepted Answer

$$(x - 4y)(x^2$ + 4xy + 16y^2$)$$

Question 8

Is $$x - 4 a $$factor of $$x^4 - 256$? If yes, what is the complete factorization?

Accepted Answer

$$(x - 4)(x + 4)(x^2$ + 16)$$

Question 9

Simplify the expression $$\frac{a^3$ - b^3$}{a^2$ - ab + b^2$}$$ and state any restrictions on the variables.

Accepted Answer

$$a - b$$, \ a 
eq b$

Question 10

Solve the linear equation $$\frac{1}{2}x + \frac{3}{1}x - \frac{3}{2}x = 6.$$

Accepted Answer

$$x = 4$$

Question 11

Given the temperature conversion formula $$F = \frac{9}{5}C + 32$$, solve for $$C in $$terms of $$F.$$

Accepted Answer

$$C = \frac{5}{9}(F - 32)$$

Question 12

Find the real solutions to the quadratic equation $$x^2 - 2x + 3 = 0$.

Accepted Answer

No real solutions

Question 13

Solve the inequality $$x \u003e 1$$ and express the solution in interval notation.

Accepted Answer

$$(1, \infty)$$

Question 14

Use the laws of exponents to simplify $$\frac{x^2$ y$$^{-2}$$}{x$$^{-3}$$ y^4$}$$, assuming all variables are positive.

Accepted Answer

Question 15

Simplify $$\sqrt[3]{8x^3$ y^6$}$$, assuming all variables are positive.

Accepted Answer

$$2x y^2$

Question 16

Rationalize the denominator of $$\frac{1}{\sqrt{x}}$$, where $$x \u003e 0.$$

Accepted Answer

$$\frac{\sqrt{x}}{x}$$

Question 17

Write $$\frac{1}{x$$^{-1}$$} as a $$single quotient with only positive exponents.

Accepted Answer

$$x$$

Question 18

Solve the rational equation $$\frac{1}{x} + \frac{1}{x - 1} = \frac{4}{x^2$ - x}$$ for $$x 
eq 0, x 
eq 1.$$

Accepted Answer

$$x = 2$$

Question 19

Solve the inequality $$x^2 - 3x$$ \u003e 0$$ and express the solution in interval notation.

Accepted Answer

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

Business Calculus Review: Algebraic Foundations and Problem-Solving Guidance

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