Question

match each product of rational expressions with its factored form.$\frac{x-2}{x^2-4}\bullet\frac{x^2-2x-8}{x+2}$$\frac{x^2-25}{x^2-2x-15}\bullet\frac{5x-5}{x^2+4x-5}$$\frac{x^2+x-6}{3x-6}\bullet\frac{x^2-2x}{2x+6}$$\frac{x^2+5x-24}{x^2+12x+32}\bullet\frac{x^2+3x-10}{x^2-6x+8}$$\frac{x^2+7x+12}{x^2-9}\bullet\frac{x-3}{x+3}$$\frac{x^2-4x}{2x+10}\bullet\frac{x^2+x-20}{3x-12}$$\frac{(x+3)(x-2)}{3(x-2)}\bullet\frac{x(x-2)}{2(x+3)}$$\frac{(x+8)(x-3)}{(x+4)(x+8)}\bullet\frac{(x+5)(x-2)}{(x-2)(x-4)}$$\frac{(x+3)(x+4)}{(x-3)(x+3)}\bullet\frac{x-3}{x+3}$$\frac{x(x-4)}{2(x+5)}\bullet\frac{(x+5)(x-4)}{3(x-4)}$$\frac{x-2}{(x-2)(x+2)}\bullet\frac{(x-4)(x+2)}{x+2}$$\frac{(x-5)(x+5)}{(x-5)(x+3)}\bullet\frac{5(x-1)}{(x-1)(x+5)}$

Explanation:

Step1: Factor first expression

$$\frac{x-2}{x^2-4} \cdot \frac{x^2-2x-8}{x+2} = \frac{x-2}{(x-2)(x+2)} \cdot \frac{(x-4)(x+2)}{x+2}$$

Step2: Factor second expression

$$\frac{x^2-25}{x^2-2x-15} \cdot \frac{5x-5}{x^2+4x-5} = \frac{(x-5)(x+5)}{(x-5)(x+3)} \cdot \frac{5(x-1)}{(x-1)(x+5)}$$

Step3: Factor third expression

$$\frac{x^2+x-6}{3x-6} \cdot \frac{x^2-2x}{2x+6} = \frac{(x+3)(x-2)}{3(x-2)} \cdot \frac{x(x-2)}{2(x+3)}$$

Step4: Factor fourth expression

$$\frac{x^2+5x-24}{x^2+12x+32} \cdot \frac{x^2+3x-10}{x^2-6x+8} = \frac{(x+8)(x-3)}{(x+4)(x+8)} \cdot \frac{(x+5)(x-2)}{(x-2)(x-4)}$$

Step5: Factor fifth expression

$$\frac{x^2+7x+12}{x^2-9} \cdot \frac{x-3}{x+3} = \frac{(x+3)(x+4)}{(x-3)(x+3)} \cdot \frac{x-3}{x+3}$$

Step6: Factor sixth expression

$$\frac{x^2-4x}{2x+10} \cdot \frac{x^2+x-20}{3x-12} = \frac{x(x-4)}{2(x+5)} \cdot \frac{(x+5)(x-4)}{3(x-4)}$$

Answer:

1. $\boldsymbol{\frac{x-2}{x^2-4} \cdot \frac{x^2-2x-8}{x+2} \longleftrightarrow \frac{x-2}{(x-2)(x+2)} \cdot \frac{(x-4)(x+2)}{x+2}}$
2. $\boldsymbol{\frac{x^2-25}{x^2-2x-15} \cdot \frac{5x-5}{x^2+4x-5} \longleftrightarrow \frac{(x-5)(x+5)}{(x-5)(x+3)} \cdot \frac{5(x-1)}{(x-1)(x+5)}}$
3. $\boldsymbol{\frac{x^2+x-6}{3x-6} \cdot \frac{x^2-2x}{2x+6} \longleftrightarrow \frac{(x+3)(x-2)}{3(x-2)} \cdot \frac{x(x-2)}{2(x+3)}}$
4. $\boldsymbol{\frac{x^2+5x-24}{x^2+12x+32} \cdot \frac{x^2+3x-10}{x^2-6x+8} \longleftrightarrow \frac{(x+8)(x-3)}{(x+4)(x+8)} \cdot \frac{(x+5)(x-2)}{(x-2)(x-4)}}$
5. $\boldsymbol{\frac{x^2+7x+12}{x^2-9} \cdot \frac{x-3}{x+3} \longleftrightarrow \frac{(x+3)(x+4)}{(x-3)(x+3)} \cdot \frac{x-3}{x+3}}$
6. $\boldsymbol{\frac{x^2-4x}{2x+10} \cdot \frac{x^2+x-20}{3x-12} \longleftrightarrow \frac{x(x-4)}{2(x+5)} \cdot \frac{(x+5)(x-4)}{3(x-4)}}$