Section 9.1: Simplifying Rational Expressions
A rational expression is a fraction in which the numerator and denominator are polynomials. Simplifying involves factoring and canceling common factors between numerator and denominator.
Example 1
Simplify: \( \frac{x^2 - 9}{x^2 - 3x} \)
Step 1: Factor numerator and denominator.
\( x^2 - 9 = (x-3)(x+3) \)
\( x^2 - 3x = x(x-3) \)
Step 2: Cancel common factor \( (x-3) \).
Result: \( \frac{x+3}{x} \), with restriction \( x \neq 0, 3 \).
Example 2
Simplify: \( \frac{2x^2 + 6x}{4x^2 + 12x} \)
Step 1: Factor numerator and denominator.
Numerator: \( 2x(x+3) \)
Denominator: \( 4x(x+3) \)
Step 2: Cancel common factors \( 2x \) and \( (x+3) \).
Result: \( \frac{1}{2} \), with restriction \( x \neq 0, -3 \).
Practice Problems
- Simplify: \( \frac{x^2 - 16}{x^2 - 4x} \)
- Simplify: \( \frac{y^2 + 5y + 6}{y^2 + 2y} \)
- Simplify: \( \frac{3x^2 - 6x}{9x - 18} \)
- Simplify: \( \frac{m^2 - m}{m^2 - 4} \)
- Simplify: \( \frac{2a^2 + 8a}{a^2 + 4a + 4} \)