Q: Simplify the rational expression $$\frac{x^2$ - 4}{x^2$ - 2x}$$ and state the excluded values.

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

Q: Multiply and simplify: $$\frac{x+3}{x^2$-4} \cdot \frac{x^2$-x-6}{x^2$+6x+9}$$, and state the excluded values.

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

Question 1

Which of the following best describes a rational expression?

Accepted Answer

A quotient of two polynomials.

Question 2

Given the rational expression $$\frac{1}{x^2$ - 9}$$, which values must be excluded from the domain?

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 3

Simplify the rational expression $$\frac{x^2$ - 4}{x^2$ - 2x}$$ and state the excluded values.

Accepted Answer

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

Question 4

A chemical reaction rate is modeled by $$R = \frac{2x}{x^2$ - 1}. $$For which values of $$x is $$the rate undefined?

Accepted Answer

$$x = 1$$ and $$x = -1$$

Question 5

Which of the following is the correct procedure to simplify $$\frac{x^2$ - 9}{x^2$ - 6x + 9}$$?

Accepted Answer

Factor both numerator and denominator, then divide by common factors.

Question 6

Multiply and simplify: $$\frac{x+3}{x^2$-4} \cdot \frac{x^2$-x-6}{x^2$+6x+9}$$, and state the excluded values.

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

A medical dosage formula is $$D = \frac{2x}{x^2$-4}. $$What values of $$x$$ must be excluded from the domain?

Accepted Answer

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

Question 9

Add the rational expressions $$\frac{2}{x-3} + \frac{5}{x-3}$$ and simplify.

Accepted Answer

$$\frac{7}{x-3}$$

Question 10

Subtract $$\frac{4}{x+2} - \frac{1}{x+2}$$ and state the excluded value.

Accepted Answer

$$\frac{3}{x+2},\ x 
eq -2$$

Question 11

Find the least common denominator (LCD) for $$\frac{1}{x^2$-4}$$ and $$\frac{1}{x^2$+2x}.$$

Accepted Answer

$$(x-2)(x+2)(x)$$

Question 12

Add $$\frac{1}{x-2}$$ and $$\frac{2}{x+3}$$ and simplify your answer.

Accepted Answer

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

Question 13

A rational expression is simplified if:

Accepted Answer

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

Question 14

Which of the following is the correct way to divide $$\frac{a}{b} by$$ $$\frac{c}{d}$$?

Accepted Answer

$$\frac{a}{b} \cdot \frac{d}{c}$$

Question 15

Simplify the complex rational expression $$\frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{x}}.$$

Accepted Answer

$$\frac{y+2x}{3y}$$

Question 16

A rational expression models the speed $$S = \frac{d}{t-2}$$, where $$d is $$distance and $$t is $$time in hours. For which value of $$t is $$the speed undefined?

Accepted Answer

$$t = 2$$

Question 17

Which step is necessary when adding $$\frac{1}{x}$$ and $$\frac{2}{x+1}$$?

Accepted Answer

Find the least common denominator.

Question 18

Simplify the complex rational expression $$\frac{\frac{x}{x+1}}{\frac{x+2}{x+1}}.$$

Accepted Answer

$$\frac{x}{x+2}$$

Rational Expressions: Fundamental Concepts in College Algebra

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