Question

Simplify the following two polynomials in the simplest form: $\frac{x^2-4}{x^2+6x+8}-\frac{x^2-11x+30}{x^2-x-20}$

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

Answer 1

Answer: (A)

$\frac{4}{x+4}$

The given expression $\frac{x^2-4}{x^2+6x+8}-\frac{x^2-11x+30}{x^2-x-20}$can be simplified as shown below:

$=\frac{x^2-4}{x^2+6x+8}-\frac{x^2-11x+30}{x^2-x-20}$

$=\frac{x^2-2^2}{x^2+4x+2x+8}-\frac{x^2-5x-6x+30}{x^2-5x+4x-20}$

$=\frac{(x-2)(x+2)}{x(x+4)+2(x+4)}-\frac{x(x-5)-6(x-5)}{x(x-5)+4(x-5)}(\because a^2-b^2=(a-b\left)\right(a+b\left)\right)$

$=\frac{(x-2)(x+2)}{(x+2)(x+4)}-\frac{(x-6)(x-5)}{(x+4)(x-5)}$

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

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

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

$=\frac{4}{x+4}$

Hence, $\frac{x^2-4}{x^2+6x+8}-\frac{x^2-11x+30}{x^2-x-20}=\frac{4}{x+4}$.