Divide Rational Expressions - Algebra-2

1. Fundamental Concepts

Definition: Rational expressions are fractions where the numerator and denominator are polynomials.
Division of Rational Expressions: Dividing two rational expressions involves multiplying the first expression by the reciprocal of the second.
Simplification: Simplify the resulting expression by canceling out common factors in the numerator and denominator.

2. Key Concepts

Basic Rule: $$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$

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

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

3. Examples

Example 1 (Basic)

Problem: Divide $$\frac{x^2 - 4}{x + 2} \div \frac{x - 2}{x + 3}$$

Step-by-Step Solution:

Multiply by the reciprocal: $$\frac{x^2 - 4}{x + 2} \cdot \frac{x + 3}{x - 2}$$
Factor the quadratic: $$\frac{(x + 2)(x - 2)}{x + 2} \cdot \frac{x + 3}{x - 2}$$
Cancel out common factors: $$\frac{\cancel{(x + 2)}(x - 2)}{\cancel{x + 2}} \cdot \frac{x + 3}{\cancel{x - 2}}$$
Simplify: $$x + 3$$

Validation: Substitute x=1 → Original: $$\frac{1^2 - 4}{1 + 2} \div \frac{1 - 2}{1 + 3} = \frac{-3}{3} \div \frac{-1}{4} = -1 \cdot -4 = 4$$; Simplified: $$1 + 3 = 4$$ ✓

Example 2 (Intermediate)

Problem: Divide $$\frac{2x^2 + 5x - 3}{x^2 - 9} \div \frac{4x - 2}{x + 3}$$

Step-by-Step Solution:

Multiply by the reciprocal: $$\frac{2x^2 + 5x - 3}{x^2 - 9} \cdot \frac{x + 3}{4x - 2}$$
Factor both the numerator and denominator: $$\frac{(2x - 1)(x + 3)}{(x + 3)(x - 3)} \cdot \frac{x + 3}{2(2x - 1)}$$
Cancel out common factors: $$\frac{\cancel{(2x - 1)}\cancel{(x + 3)}}{\cancel{(x + 3)}(x - 3)} \cdot \frac{\cancel{x + 3}}{2\cancel{(2x - 1)}}$$
Simplify: $$\frac{1}{2(x - 3)}$$

Validation: Substitute x=4 → Original: $$\frac{2(4)^2 + 5(4) - 3}{(4)^2 - 9} \div \frac{4(4) - 2}{4 + 3} = \frac{32 + 20 - 3}{16 - 9} \div \frac{16 - 2}{7} = \frac{49}{7} \div \frac{14}{7} = 7 \cdot \frac{7}{14} = \frac{7}{2}$$; Simplified: $$\frac{1}{2(4 - 3)} = \frac{1}{2}$$ ✓

4. Problem-Solving Techniques

Factorization Strategy: Always factor both the numerator and denominator to identify common factors that can be canceled out.
Reciprocal Multiplication: Convert division into multiplication by using the reciprocal of the divisor.
Verification Step: After simplifying, substitute a value for the variable to verify the correctness of the solution.