Section 4.3: Rational Expressions Simplification
This section teaches how to simplify rational expressions by factoring numerators and denominators and reducing common factors.
Example 1: Simplifying a Rational Expression
Simplify \( \frac{x^2 - 9}{x^2 - 6x + 9} \).
Step 1: Factor numerator: \( x^2 - 9 = (x-3)(x+3) \)
Step 2: Factor denominator: \( x^2 - 6x + 9 = (x-3)^2 \)
Step 3: Cancel common factor: \( \frac{x+3}{x-3} \)
Example 2: Simplifying a Complex Rational Expression
Simplify \( \frac{x^2 - 4x + 4}{x^2 - 2x} \).
Step 1: Factor numerator: \( x^2 - 4x + 4 = (x-2)^2 \)
Step 2: Factor denominator: \( x^2 - 2x = x(x-2) \)
Step 3: Cancel common factor: \( \frac{x-2}{x} \)
Practice Problems
- Simplify \( \frac{x^2 - 16}{x^2 - 8x + 16} \)
- Simplify \( \frac{2x^2 + 6x}{4x^2 + 8x} \)
- Simplify \( \frac{x^2 + 5x + 6}{x^2 + 3x} \)
- Simplify \( \frac{x^2 - 9x + 20}{x^2 - 4} \)
- Simplify \( \frac{3x^2 - 12}{6x^2 - 24} \)