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.

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