Question

multiply the rational expressions. write your answer in simplest form. $\frac{3x - 15}{5x^{2}-25x}cdot\frac{3x^{2}+12x}{3x^{2}-75}$
steps: 1. factor all terms 2. reduce common terms 3. multiply & write in simplest factored form

Explanation:

Step1: Factor the expressions

• Factor $3x - 15$: $3x-15 = 3(x - 5)$
• Factor $5x^{2}-25x$: $5x^{2}-25x=5x(x - 5)$
• Factor $3x^{2}+12x$: $3x^{2}+12x = 3x(x + 4)$
• Factor $3x^{2}-75$: $3x^{2}-75=3(x^{2}-25)=3(x + 5)(x - 5)$

The original expression $\frac{3x - 15}{5x^{2}-25x}\cdot\frac{3x^{2}+12x}{3x^{2}-75}$ becomes $\frac{3(x - 5)}{5x(x - 5)}\cdot\frac{3x(x + 4)}{3(x + 5)(x - 5)}$

Step2: Cancel out common terms

Cancel out the common factors:

• Cancel out the common factor of $(x - 5)$ in the first - fraction's numerator and denominator, and also cancel out the common factor of $3x$ between the two fractions.

We get $\frac{3}{5x}\cdot\frac{(x + 4)}{(x + 5)(x - 5)}$

Step3: Multiply the remaining terms

Multiply the numerators and denominators: $\frac{3(x + 4)}{5x(x + 5)(x - 5)}$

Answer:

$\frac{3(x + 4)}{5x(x + 5)(x - 5)}$