Algebra 101 - Unit 8

Lesson 46: Simplifying Rational Expressions

Rational expressions are fractions in which the numerator and/or denominator are polynomials. Simplifying them involves factoring and reducing common factors.

Steps to Simplify Rational Expressions

1. Factor the numerator and denominator completely.
2. Identify and cancel any common factors between numerator and denominator.
3. Write the simplified expression.
4. State any restrictions on the variable (values that make the denominator zero).

Examples

Example 1: Simplify f(x) = (x² − 9)/(x² − 6x + 9)

• Factor numerator: x² − 9 = (x − 3)(x + 3)
• Factor denominator: x² − 6x + 9 = (x − 3)(x − 3)
• Cancel common factor: (x + 3)/(x − 3)
• Restriction: x ≠ 3

Example 2: Simplify g(x) = (2x² − 8)/(4x² − 16)

• Factor numerator: 2(x² − 4) = 2(x − 2)(x + 2)
• Factor denominator: 4(x² − 4) = 4(x − 2)(x + 2)
• Cancel common factors: 2/4 = 1/2
• Simplified: 1/2
• Restrictions: x ≠ ±2

Practice Problems

1. Simplify: (x² − 16)/(x² − 4x)
2. Simplify: (3x² − 12)/(6x² − 24x + 24)
3. Simplify: (x² + 5x + 6)/(x² + 3x)
4. Simplify: (4x² − 9)/(2x² − 3x)
5. Simplify: (x³ − x)/(x² − 1)