1. **Problem Statement:** We will review algebraic fractions, including simplification, finding common denominators, addition, subtraction, multiplication, and division. 2. **Simplification:** To simplify an algebraic fraction, factor numerator and denominator and cancel common factors. Example: Simplify $$\frac{x^2 - 9}{x^2 - 6x + 9}$$. Factor numerator: $$x^2 - 9 = (x-3)(x+3)$$. Factor denominator: $$x^2 - 6x + 9 = (x-3)^2$$. Cancel common factor $x-3$: $$\frac{(x-3)(x+3)}{(x-3)(x-3)} = \frac{x+3}{x-3}$$ (with $x \neq 3$). 3. **Finding Common Denominators:** To add or subtract fractions, find the least common denominator (LCD), which is the least common multiple of denominators. Example: Add $$\frac{1}{x} + \frac{1}{x+2}$$. LCD is $$x(x+2)$$. Rewrite fractions: $$\frac{1}{x} = \frac{x+2}{x(x+2)}$$ and $$\frac{1}{x+2} = \frac{x}{x(x+2)}$$. Add: $$\frac{x+2}{x(x+2)} + \frac{x}{x(x+2)} = \frac{2x+2}{x(x+2)} = \frac{2(x+1)}{x(x+2)}$$. 4. **Addition and Subtraction:** Use the common denominator and combine numerators. 5. **Multiplication:** Multiply numerators and denominators directly, then simplify. Example: Multiply $$\frac{x}{x+1} \times \frac{x-1}{x}$$. Multiply: $$\frac{x(x-1)}{(x+1)x} = \frac{x-1}{x+1}$$ (cancel $x$). 6. **Division:** Multiply by the reciprocal of the divisor. Example: Divide $$\frac{x}{x+1} \div \frac{x-1}{x}$$. Rewrite as multiplication: $$\frac{x}{x+1} \times \frac{x}{x-1} = \frac{x^2}{(x+1)(x-1)}$$. **Summary:** - Simplify by factoring and canceling. - Find LCD for addition/subtraction. - Multiply numerators and denominators. - Divide by multiplying by reciprocal. These rules help manipulate algebraic fractions effectively.