Question

Write as equivalent rational expression with LCD$(x-4)(x+2)(x+3)$ of$\frac{5}{x^2-2x-8},\frac{2x}{x^2-x-12}$.

Short answer

The rational expression is$\frac{5x+15}{(x-4)(x+2)(x+3)},\frac{2x^2+4x}{(x-4)(x+3)(x+2)}$.

Step-by-step solution

Given Information

The given expression is$\frac{5}{x^2-2x-8},\frac{2x}{x^2-x-12}$.

The LCD is$(x-4)(x+2)(x+3)$.

Factor the denominator of both expressions.

The factored form of first denominator.

$x^2-2x-8=(x-4)(x+2)$

The factored form of second denominator.

$x^2-x-12=(x-4)(x+3)$

The fractions in factored form is$\frac{5}{(x-4)(x+2)},\frac{2x}{(x-4)(x+3)}$.

Multiply each denominator by the 'missing' factor and multiplyeach numerator by the same factor.

$\frac{5(x+3)}{(x-4)(x+2)(x+3)},\frac{2x(x+2)}{(x-4)(x+3)(x+2)}$

Simplify the numerators.

Multiply the numerator using distributive property.

$\frac{5x+15}{(x-4)(x+2)(x+3)},\frac{2x^2+4x}{(x-4)(x+3)(x+2)}$.