Master Rational Expressions: Simplify, Multiply, and Divide with Confidence

Question 1

Which of these is an invalid step in simplifying a rational expression?

Accepted Answer

Adding or subtracting fractions

Answer

Factoring the numerator and denominator

Answer

Canceling common factors

Answer

Multiplying the numerator and denominator by the same expression

Question 2

Identify the domain restriction for the rational expression (x² - 1)/(x - 1)?

Accepted Answer

x = 1

Answer

x = 0

Answer

x = 2

Answer

x = -1

Question 3

Determine the vertical asymptote for the rational function f(x) = ((x² - 9)/(x - 3))

Accepted Answer

x = 3

Answer

x = 0

Answer

x = -3

Answer

x = 2

Question 4

Find the least common multiple of the denominators of the following rational expressions:

$$rac{2}{x - 1} 	ext{ and } rac{3}{x^2 + x - 2}$$

Accepted Answer

$$(x - 1)(x^2 + x - 2)$$

Answer

$$x^2 + x - 2$$

Answer

$$x - 1$$

Answer

$$x^2 + 2x - 1$$

Question 5

Find the least common denominator (LCD) of the rational expressions: (2/x - 3) and (5/2x + 6)

Accepted Answer

2x(x + 3)

Answer

2x + 6

Answer

2x

Answer

x - 3

Question 6

Determine if the rational expression (x² - 4) / (x - 2) is undefined for any value of x.

Accepted Answer

x = 2

Answer

x = 4

Answer

x = 0

Answer

It is defined for all values of x

Question 7

Find the domain of the rational expression: (2x + 1) / (x² - 4)

Accepted Answer

x ∈ R, except x = -2, 2

Answer

x ∈ R

Answer

x < 0

Answer

x > 0

Question 8

Identify the vertical asymptote of the rational expression: (x + 3) / (x - 1)

Accepted Answer

x = 1

Answer

y = 3

Answer

y = 1

Answer

x = 3

Question 9

Simplify the rational expression: (x² - 4) / (x + 2)

Accepted Answer

x - 2

Answer

x² - 2

Answer

(x - 2)(x + 2)

Answer

x + 2

Question 10

What is the simplified product of (x + 3) / (x - 1) and (x - 1) / (x² - 9)?

Accepted Answer

1 / (x - 3)

Answer

x + 3 / (x - 3)

Answer

x + 3 / (x² - 9)

Answer

x - 1 / (x + 3)

Question 11

Which expression is equivalent to (2x + 4) / (x² - 4)?

Accepted Answer

2 / (x - 2)

Answer

2x / (x + 2)

Answer

2x + 4 / (x + 2)

Answer

2 / (x + 2)

Question 12

What are the excluded values for the rational expression (x² - 1) / (x² - 4x + 3)?

Accepted Answer

x = 1, x = 3

Answer

x = -1, x = -3

Answer

x = 1, x = -3

Answer

x = -1, x = 3

Question 13

Simplify the complex rational expression: ((1/x) + (1/y)) / (1/x - 1/y)

Accepted Answer

(x + y) / (x - y)

Answer

x / y

Answer

y / x

Answer

(x - y) / (x + y)

Question 14

Simplify the rational expression: (x² - 4) / (x + 2)

Accepted Answer

x - 2

Answer

x² - 2

Answer

(x - 2)(x + 2)

Answer

x + 2

Question 15

What is the least common multiple (LCM) of the denominators of the expressions: 1/x, 2/(x-1), and 3/(x² - 1)?

Accepted Answer

x(x-1)(x+1)

Answer

x(x-1)

Answer

x² - 1

Answer

(x-1)(x+1)

Question 16

Add the rational expressions: (x+1)/x + (x-1)/(x+2)

Accepted Answer

(2x² + 4x + 2)/(x(x+2))

Answer

(2x² + 2x + 2)/(x(x+2))

Answer

(x² + 3x + 1)/(x(x+2))

Answer

(2x² + 2)/(x(x+2))

Question 17

Subtract the rational expressions: (x² + 3x) / (x² - 1) - (2x) / (x+1)

Accepted Answer

(x² + x) / (x² - 1)

Answer

(x² + 3x - 2x) / (x² - 1)

Answer

(x² + x) / (x-1)

Answer

(x² + 3x - 2x) / (x-1)

Question 18

Multiply the rational expressions: (x² - 9) / (x + 3) * (x + 1) / (x - 3)

Accepted Answer

x + 1

Answer

(x² - 9)(x + 1) / (x + 3)(x - 3)

Answer

x - 1

Answer

(x - 3)(x + 1) / (x + 3)

Question 19

For what value(s) of x is the rational expression (x^2 - 4) / (x - 2) undefined?

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 0

Answer

x = 2 and x = -2

Question 20

Simplify the complex rational expression: (1/x + 1/y) / (1/x - 1/y)

Accepted Answer

(x + y) / (x - y)

Answer

1

Answer

x/y

Answer

(x - y) / (x + y)

Question 21

Solve the following equation for x: (x + 1) / (x - 2) = 3

Accepted Answer

x = 7/2

Answer

x = 2

Answer

x = -1

Answer

x = -7/2

Question 22

Which of the following is a simplified form of the rational expression (x^2 + 2x + 1) / (x^2 - 1)?

Accepted Answer

(x + 1) / (x - 1)

Answer

x - 1

Answer

(x + 1)^2 / (x - 1)

Answer

x + 1

Question 23

Simplify the rational expression: (x² - 4) / (x + 2)

Accepted Answer

x - 2

Answer

x + 2

Answer

x + 4

Answer

x² - 2

Question 24

What is the product of (x + 3) / (x - 1) and (x² - 1) / (x² + 6x + 9)?

Accepted Answer

x + 1

Answer

(x + 3) / (x - 1)

Answer

(x² - 1) / (x² + 6x + 9)

Answer

(x + 1) / (x + 3)

Question 25

Divide the rational expression (x² - 9) / (x + 3) by (x - 3) / (x² + 6x + 9).

Accepted Answer

(x + 3)²

Answer

(x - 3) / (x + 3)

Answer

(x + 3) / (x - 3)

Answer

x - 3

Question 26

Find the least common multiple (LCM) of (x² - 4) and (x² + 2x).

Accepted Answer

x(x + 2)(x - 2)

Answer

x² - 4

Answer

x² + 2x

Answer

x(x + 2)

Question 27

Simplify the expression: (2 / (x + 1)) + (3 / (x - 1)).

Accepted Answer

(5x + 1) / (x² - 1)

Answer

(5x - 1) / (x² - 1)

Answer

5 / (2x)

Answer

5 / (x² - 1)

Question 28

Subtract the rational expression (x / (x + 2)) - (1 / (x - 2)).

Accepted Answer

(x² - 3x - 2) / (x² - 4)

Answer

(x² - 3x + 2) / (x² - 4)

Answer

(x² + 3x - 2) / (x² - 4)

Question 29

What is the excluded value for the rational expression (x - 5) / (x² - 25)?

Accepted Answer

x = 5

Answer

x = 0

Answer

x = -5

Answer

x = 25

Question 30

Which of the following is NOT a rational expression?

Accepted Answer

√(x + 1)

Answer

(x + 2) / (x - 3)

Answer

3 / (x² + 4)

Answer

x² - 1

Question 31

Solve the rational equation: 2 / (x - 1) = 1 / (x + 2).

Accepted Answer

x = 0

Answer

x = -2

Answer

x = 1

Answer

x = 2

Question 32

Simplify the complex rational expression: ((x + 1) / (x - 1)) / ((x - 1) / (x + 1)).

Accepted Answer

(x + 1)² / (x - 1)²

Answer

(x + 1) / (x - 1)

Answer

(x - 1) / (x + 1)

Answer

1