Question

Simplify the following two polynomials in the simplest form: $\frac{x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}$

• A. $\frac{2(x+4)}{x+3}$
• B. $\frac{2(x-4)}{x+3}$
• C. $\frac{2(x+4)}{x-3}$
• D. None of these

Answer 1

Answer: (A)

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

The given expression $\frac{x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}$can be simplified as shown below:

$=\frac{x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}$

$=\frac{x^2-3x+2x-6}{x^2-3^2}+\frac{x^2+6x-4x-24}{x^2-4x+3x-12}$

$=\frac{x(x-3)+2(x-3)}{(x-3)(x+3)}+\frac{x(x+6)-4(x+6)}{x(x-4)+3(x-4)}(\because a^2-b^2=(a-b\left)\right(a+b\left)\right)$

$=\frac{(x+2)(x-3)}{(x-3)(x+3)}+\frac{(x-4)(x+6)}{(x+3)(x-4)}$

$=\frac{x+2}{x+3}+\frac{x+6}{x+3}$

$=\frac{x+2+x+6}{x+3}$

$=\frac{2x+8}{x+3}$

$=\frac{2(x+4)}{x+3}$

Hence, $\frac{x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}=\frac{2(x+4)}{x+3}$.