Question

Simplify the following two polynomials in the simplest form: $\frac{x+2}{x^2+3x+2}+\frac{x-3}{x^2-2x-3}$

• A. $\frac{2}{x+1}$
• B. $\frac{2}{x-1}$
• C. $\frac{2}{x+2}$
• D. None of these

Answer 1

Answer: (A)

$\frac{2}{x+1}$

The given expression $\frac{x+2}{x^2+3x+2}+\frac{x-3}{x^2-2x-3}$can be simplified as shown below:

$=\frac{x+2}{x^2+3x+2}+\frac{x-3}{x^2-2x-3}$

$=\frac{x+2}{x^2+2x+x+2}+\frac{x-3}{x^2-3x+x-3}$

$=\frac{x+2}{x(x+2)+1(x+2)}+\frac{x-3}{x(x-3)+1(x-3)}$

$=\frac{x+2}{(x+1)(x+2)}+\frac{x-3}{(x+1)(x-3)}$

$=\frac{1}{(x+1)}+\frac{1}{(x+1)}$

$=\frac{1+1}{x+1}$

$=\frac{2}{x+1}$

Hence, $\frac{x+2}{x^2+3x+2}+\frac{x-3}{x^2-2x-3}=\frac{2}{x+1}$.