Understanding Rational Expressions

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Algebra II › Understanding Rational Expressions

Questions 1 - 10

1

Simplify the expression:

$\frac{5}{x+ 6} + \frac{2}{x^{2}+ 4x-12}$

$\frac{5x-8}{ x^{2}+ 4x-12}$

$\frac{5x-12}{x^{2}+ 4x-12}$

$\frac{5x }{x^{2}+ 4x-12}$

$\frac{5x+2}{x-2}$

$\frac{5x-2}{x-2}$

Explanation

Factor the second denominator, then simplify:

$\frac{5}{x+ 6} + \frac{2}{x^{2}+ 4x-12}$

$=\frac{5}{x+ 6} + \frac{2}{(x-2)(x+6)}$

$=\frac{5 (x-2)}{ (x-2)(x+6)} + \frac{2}{(x-2)(x+6)}$

$=\frac{5x-10}{ (x-2)(x+6)} + \frac{2}{(x-2)(x+6)}$

$=\frac{5x-10 + 2}{ (x-2)(x+6)}$

$=\frac{5x-8}{ (x-2)(x+6)}$

$=\frac{5x-8}{x^{2}+ 4x-12}$

2

Simplify the expression:

$\frac{5}{x+ 6} + \frac{2}{x^{2}+ 4x-12}$

$\frac{5x-8}{ x^{2}+ 4x-12}$

$\frac{5x-12}{x^{2}+ 4x-12}$

$\frac{5x }{x^{2}+ 4x-12}$

$\frac{5x+2}{x-2}$

$\frac{5x-2}{x-2}$

Explanation

Factor the second denominator, then simplify:

$\frac{5}{x+ 6} + \frac{2}{x^{2}+ 4x-12}$

$=\frac{5}{x+ 6} + \frac{2}{(x-2)(x+6)}$

$=\frac{5 (x-2)}{ (x-2)(x+6)} + \frac{2}{(x-2)(x+6)}$

$=\frac{5x-10}{ (x-2)(x+6)} + \frac{2}{(x-2)(x+6)}$

$=\frac{5x-10 + 2}{ (x-2)(x+6)}$

$=\frac{5x-8}{ (x-2)(x+6)}$

$=\frac{5x-8}{x^{2}+ 4x-12}$

3

Which value of makes the following expression undefined?

$\frac{x+1}{x-1}$

$\text{All real numbers}$

$\text{Cannot be determined}$

Explanation

A rational expression is undefined when the denominator is zero.

The denominator is zero when .

4

Which value of makes the following expression undefined?

$\frac{x+1}{x-1}$

$\text{All real numbers}$

$\text{Cannot be determined}$

Explanation

A rational expression is undefined when the denominator is zero.

The denominator is zero when .

5

$\frac{(x^{2}-1x-6)}{x+3}+\frac{(x^{2}-6x+9)}{x+2}$

What is the least common denominator of the above expression?

$(x+3)^{2}+(x-2)$

None of these answer choices

Explanation

The least common denominator is the least common multiple of the denominators of a set of fractions.

Simply multiply the two denominators together to find the LCD:

6

$\frac{(x^{2}-1x-6)}{x+3}+\frac{(x^{2}-6x+9)}{x+2}$

What is the least common denominator of the above expression?

$(x+3)^{2}+(x-2)$

None of these answer choices

Explanation

The least common denominator is the least common multiple of the denominators of a set of fractions.

Simply multiply the two denominators together to find the LCD:

7

Find the least common denominator of the following fractions:

$\frac{6}{7} , \frac{4}{3} , \frac{5}{9}$

Explanation

The denominators are 7, 3, and 9. We have to find the common multiple of 7, 3, and 9.

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63

Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

The least common multiple of the 3 denominators is 63.

8

Find the least common denominator of the following fractions:

$\frac{6}{7} , \frac{4}{3} , \frac{5}{9}$

Explanation

The denominators are 7, 3, and 9. We have to find the common multiple of 7, 3, and 9.

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63

Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

The least common multiple of the 3 denominators is 63.

9

What is the least common denominator needed to continue with the following problem?

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

$\frac{x+12}{x^2-x-6}$

$-\frac{x+12}{x^2-x-6}$

Explanation

In order to continue to simplify this problem, we will need to multiply both denominators together in order to simplify the numerators.

Multiply both of the denominators by using the FOIL method.

Simplify this expression.

The least common denominator required is:

Be careful not to continue and solve the problem!

The answer is:

10

What is the least common denominator needed to continue with the following problem?

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

$\frac{x+12}{x^2-x-6}$

$-\frac{x+12}{x^2-x-6}$

Explanation

In order to continue to simplify this problem, we will need to multiply both denominators together in order to simplify the numerators.

Multiply both of the denominators by using the FOIL method.

Simplify this expression.

The least common denominator required is:

Be careful not to continue and solve the problem!

The answer is:

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