Q: Simplify and state the restrictions: $$\frac{x^2$-4}{x^2$+2x}$$

$$\frac{x-2}{x}$$, $$x \neq 0, x \neq -2$$

Question 1

Which of the following is the correct definition of a rational expression?

Accepted Answer

An expression that can be written as the ratio of two polynomials, where the denominator is not zero.

Question 2

Simplify the following rational expression: $$\frac{x^2$ - 9}{x^2$ - 6x + 9}$$

Accepted Answer

$$\frac{x+3}{x-3}$$

Question 3

Which value(s) of $$x$$ make the rational expression $$\frac{2x+5}{x^2$-4}$$ undefined?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 4

A chemist is mixing two solutions. The concentration of the first solution is represented by $$\frac{3x}{x+2}$$ and the second by $$\frac{2x}{x-2}. $$What is the sum of the concentrations?

Accepted Answer

$$\frac{3x(x-2) + 2x(x+2)}{(x+2)(x-2)}$$

Question 5

Simplify and state the restrictions: $$\frac{x^2$-4}{x^2$+2x}$$

Accepted Answer

$$\frac{x-2}{x}$$, $$x 
eq 0, x 
eq -2$$

Question 6

Multiply and simplify: $$\frac{2x}{x^2$-9} 	imes \frac{x+3}{4}$$

Accepted Answer

$$\frac{2x}{4(x-3)}$$

Question 7

Divide and simplify: $$\frac{x^2$-1}{x+2} \div \frac{x-1}{x-2}$$

Accepted Answer

$$\frac{(x^2$-1)(x-2)}{(x+2)(x-1)}$$

Question 8

A medical researcher is analyzing the rate of a drug's absorption, modeled by $$\frac{5t}{t^2$+4t}. $$What is the domain of this function?

Accepted Answer

All real numbers except $$t = 0$$ and $$t = -4$$

Question 9

Simplify: $$\frac{3}{x} + \frac{2}{x+1}$$

Accepted Answer

$$\frac{3(x+1)+2x}{x(x+1)}$$

Question 10

If $$\frac{a}{b} = \frac{c}{d}$$ and $$b, d 
eq 0$$, which of the following is true?

Accepted Answer

$$a \cdot d = b \cdot c$$

Question 11

A company’s profit is modeled by $$P(x) = \frac{2x^2$ + 3x}{x}$$ for $$x 
eq 0. $$Simplify $$P(x).$$

Accepted Answer

$$2x + 3$$

Question 12

Which of the following is equivalent to $$\frac{x^2$-16}{x^2$-8x+16}$$?

Accepted Answer

$$\frac{(x-4)(x+4)}{(x-4)^2$}$$

Question 13

Subtract and simplify: $$\frac{5}{x-2} - \frac{3}{x+2}$$

Accepted Answer

$$\frac{5(x+2)-3(x-2)}{(x-2)(x+2)}$$

Question 14

If $$f(x) = \frac{1}{x-1}$$ and $$g(x) = \frac{2}{x+1}$$, what is $$f(x) + g(x)$$?

Accepted Answer

$$\frac{(x+1)+2(x-1)}{(x-1)(x+1)}$$

Question 15

A rational expression is undefined for which of the following values of $$x in$$ $$\frac{4x}{x^2$-5x}$$?

Accepted Answer

$$x = 0$$ and $$x = 5$$

Question 16

Simplify: $$\frac{x^2$-25}{x^2$+10x+25}$$

Accepted Answer

$$\frac{(x-5)(x+5)}{(x+5)^2$}$$

Question 17

A rational function models the speed of a car as $$S(t) = \frac{60}{t-1}$$, where $$t is $$time in hours. For which value of $$t is $$the speed undefined?

Accepted Answer

$$t = 1$$

Question 18

Which of the following is the correct process for dividing two rational expressions?

Accepted Answer

Multiply the first expression by the reciprocal of the second.

Question 19

If $$\frac{a}{b} = \frac{c}{d}$$, what is the cross-multiplication result?

Accepted Answer

$$ad = bc$$

Question 20

A rational expression is simplified if:

Accepted Answer

The numerator and denominator have no common factors other than 1.

Rational Expressions: Homework Overview and Study Guide

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