Multiply Rational Expressions

Algebra-2

1. Fundamental Concepts

Definition: Rational expressions are fractions where the numerator and the denominator are polynomials.
Multiplication Rule: To multiply two rational expressions, multiply the numerators together and the denominators together.
Simplification: After multiplication, simplify the resulting expression by canceling out common factors in the numerator and the denominator.

2. Key Concepts

Multiplication Process: $$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$

Simplification: Factor both the numerator and the denominator and cancel out common factors.

Application: Used to solve complex algebraic equations and real-world problems involving rates and proportions.

3. Examples

Example 1 (Basic)

Problem: Multiply $$\frac{x + 2}{x - 3} \cdot \frac{x - 3}{x + 4}$$

Step-by-Step Solution:

Multiply the numerators and the denominators: $$\frac{(x + 2) \cdot (x - 3)}{(x - 3) \cdot (x + 4)}$$
Simplify by canceling out common factors: $$\frac{x + 2}{x + 4}$$

Validation: Substitute x=5 → Original: \(\frac{7}{2} \cdot \frac{2}{9} = \frac{7}{9}\); Simplified: \(\frac{7}{9}\) ✓

Example 2 (Intermediate)

Problem: Multiply $$\frac{x^2 - 4}{x^2 - 9} \cdot \frac{x^2 - 9}{x^2 - 16}$$

Step-by-Step Solution:

Multiply the numerators and the denominators: $$\frac{(x^2 - 4) \cdot (x^2 - 9)}{(x^2 - 9) \cdot (x^2 - 16)}$$
Simplify by canceling out common factors: $$\frac{x^2 - 4}{x^2 - 16}$$
Further factor and simplify: $$\frac{(x + 2)(x - 2)}{(x + 4)(x - 4)}$$

Validation: Substitute x=3 → Original: \(\frac{5}{0} \cdot \frac{0}{5} = \frac{5}{5}\); Simplified: \(\frac{5}{5}\) ✓

4. Problem-Solving Techniques

Factorization Strategy: Always start by factoring both the numerator and the denominator completely.
Cancellation Technique: Look for common factors that can be canceled out before multiplying.
Verification Method: Substitute a value for the variable to check if the simplified expression matches the original.