Full Prep Academy

Overview

Multiplying and dividing rational expressions involves working with fractions where both the numerator and denominator are polynomials. These operations follow similar steps to working with simple fractions but with the added complexity of factoring the polynomials involved.

Multiplying Rational Expressions

To multiply two rational expressions:

1. Factor both the numerator and denominator of each expression.
2. Multiply the numerators together and the denominators together.
3. Cancel out any common factors between the numerator and the denominator.
4. Simplify the expression.

Example 1: Multiplying Rational Expressions

Consider the following multiplication problem:

\( \frac{x^2 + 5x + 6}{x^2 - 4} \times \frac{x^2 - 9}{x^2 + 2x} \)

Step 1: Factor both the numerators and denominators:

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

Step 2: Multiply the numerators and denominators:

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

Step 3: Cancel out common factors:

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

Step 4: Simplify the remaining expression:

\( \frac{x^2 - 9}{x(x - 2)} \)

Dividing Rational Expressions

To divide two rational expressions, follow these steps:

1. Factor both the numerator and denominator of each expression.
2. Invert the second expression (i.e., flip the numerator and denominator).
3. Multiply the first expression by the inverted second expression.
4. Cancel out any common factors between the numerator and denominator.
5. Simplify the expression.

Example 2: Dividing Rational Expressions

Consider the following division problem:

\( \frac{x^2 - 4}{x^2 - x - 6} \div \frac{x^2 - 16}{x^2 - 5x + 6} \)

Step 1: Factor both the numerators and denominators:

\( \frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} \div \frac{(x - 4)(x + 4)}{(x - 3)(x - 2)} \)

Step 2: Invert the second expression and multiply:

\( \frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} \times \frac{(x - 3)(x - 2)}{(x - 4)(x + 4)} \)

Step 3: Cancel out common factors:

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

Step 4: Simplify the remaining expression:

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

Practice Questions

1. Question 1: Multiply the following rational expressions:

   \( \frac{x^2 - 4}{x^2 - 9} \times \frac{x^2 - 1}{x^2 + 2x} \)

   Solution

   Step 1: Factor both the numerators and denominators:

   \( \frac{(x - 2)(x + 2)}{(x - 3)(x + 3)} \times \frac{(x - 1)(x + 1)}{x(x + 2)} \)

   Step 2: Multiply the numerators and denominators:

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

   Step 3: Cancel out common factors:

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

   So, the simplified expression is:

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

2. Question 2: Divide the following rational expressions:

   \( \frac{x^2 - 4}{x^2 - 2x - 8} \div \frac{x^2 - 1}{x^2 - x - 6} \)

   Solution

   Step 1: Factor both the numerators and denominators:

   \( \frac{(x - 2)(x + 2)}{(x - 4)(x + 2)} \div \frac{(x - 1)(x + 1)}{(x - 3)(x + 2)} \)

   Step 2: Invert the second expression and multiply:

   \( \frac{(x - 2)(x + 2)}{(x - 4)(x + 2)} \times \frac{(x - 3)(x + 2)}{(x - 1)(x + 1)} \)

   Step 3: Cancel out common factors:

   \( \frac{(x - 2)(x - 3)}{(x - 4)(x - 1)(x + 1)} \)

   So, the simplified expression is:

   \( \frac{(x - 2)(x - 3)}{(x - 4)(x - 1)(x + 1)} \)

3. Question 3: Multiply the following rational expressions:

   \( \frac{2x^2 - 4x}{x^2 - 5x + 6} \times \frac{x^2 - 9}{x^2 - 3x} \)

   Solution

   Step 1: Factor both the numerators and denominators:

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

   Step 2: Multiply the numerators and denominators:

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

   Step 3: Cancel out common factors:

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

   So, the simplified expression is:

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