Rational Expression Division 38F7B6

1. **State the problem:** Find the quotient of the given rational algebraic expressions. 2. **Recall the division rule for rational expressions:** $$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{A \cdot D}{B \cdot C}$$ where $B$, $C$, and $D$ are nonzero. 3. **Example 1:** Given: $$\frac{a}{a-b} \div \frac{b}{a-b}$$ Step 1: Take the reciprocal of the divisor: $$\frac{a}{a-b} \cdot \frac{a-b}{b}$$ Step 2: Multiply numerators and denominators: $$\frac{a \cdot (a-b)}{(a-b) \cdot b}$$ Step 3: Cancel the common factor $a-b$: $$\frac{a \cdot \cancel{(a-b)}}{\cancel{(a-b)} \cdot b} = \frac{a}{b}$$ 4. **Example 2:** Given: $$\frac{x^2 - 6x + 9}{4x - 12} \div (x - 3)$$ Step 1: Rewrite division as multiplication by reciprocal: $$\frac{x^2 - 6x + 9}{4x - 12} \cdot \frac{1}{x - 3}$$ Step 2: Factor numerator and denominator where possible: $$\frac{(x - 3)(x - 3)}{4(x - 3)} \cdot \frac{1}{x - 3}$$ Step 3: Multiply numerators and denominators: $$\frac{(x - 3)(x - 3) \cdot 1}{4(x - 3)(x - 3)}$$ Step 4: Cancel common factors $\cancel{(x - 3)}$ and $\cancel{(x - 3)}$: $$\frac{\cancel{(x - 3)} \cdot \cancel{(x - 3)}}{4 \cdot \cancel{(x - 3)} \cdot \cancel{(x - 3)}} = \frac{1}{4}$$ 5. **Summary:** To divide rational algebraic expressions: - Take the reciprocal of the divisor. - Change division to multiplication. - Multiply numerators and denominators. - Factor and cancel common factors. 6. **Final answer for Example 1:** $$\frac{a}{b}$$ 7. **Final answer for Example 2:** $$\frac{1}{4}$$ This method applies to all rational algebraic expression division problems.