Question

divide the rational expressions. write your answer in simplest form. $\frac{x + 2}{3x^{2}}div\frac{6x + 12}{3x^{3}-15x^{2}}$ $\frac{x + 2}{3x^{2}}div\frac{6x + 12}{3x^{3}-15x^{2}}=\frac{x - 5}{6}$ $\frac{x + 2}{3x^{2}}div\frac{6x + 12}{3x^{3}-15x^{2}}=\frac{3x^{2}(3x^{3}-15x^{2})}{(x + 2)(6x + 12)}$ $\frac{x + 2}{3x^{2}}div\frac{6x + 12}{3x^{3}-15x^{2}}=\frac{(x + 2)(6x + 12)}{3x^{2}(3x^{3}-15x^{2})}$ $\frac{x + 2}{3x^{2}}div\frac{6x + 12}{3x^{3}-15x^{2}}=\frac{6}{x - 5}$ steps: 1. multiply by reciprocal 2. factor all terms 3. reduce common terms 3. write in simplest factored form

Explanation:

Step1: Multiply by reciprocal

$\frac{x + 2}{3x^{2}}\div\frac{6x + 12}{3x^{3}-15x^{2}}=\frac{x + 2}{3x^{2}}\times\frac{3x^{3}-15x^{2}}{6x + 12}$

Step2: Factor all terms

Factor out common factors: $6x + 12=6(x + 2)$, $3x^{3}-15x^{2}=3x^{2}(x - 5)$.
So the expression becomes $\frac{x + 2}{3x^{2}}\times\frac{3x^{2}(x - 5)}{6(x + 2)}$

Step3: Reduce common terms

Cancel out the common factors $(x + 2)$ and $3x^{2}$ in the numerator and denominator.
We get $\frac{x - 5}{6}$

Answer:

$\frac{x - 5}{6}$