Question

Simplify the following polynomials in the simplest form: $\frac{2x+5}{x+1}+\frac{x^2+1}{x^2-1}-\left(\frac{3x-2}{x-1}\right)$

• A. $\frac{2}{x+1}$
• B. $\frac{2}{x-1}$
• C. $\frac{4}{x+1}$
• D. None of these

Answer 1

Answer: (A)

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

The given expression $[\frac{2x+5}{x+1}+\frac{x^2+1}{x^2-1}]-\left(\frac{3x-2}{x-1}\right)$can be simplified as shown below:

$=[\frac{2x+5}{x+1}+\frac{x^2+1}{x^2-1}]-\left(\frac{3x-2}{x-1}\right)$

$=[\frac{2x+5}{x+1}+\frac{x^2+1}{(x-1)(x+1)}]-\left(\frac{3x-2}{x-1}\right)(\because a^2-b^2=(a-b\left)\right(a+b\left)\right)$

$=[\frac{(2x+5)(x-1)}{(x+1)(x-1)}+\frac{x^2+1}{(x-1)(x+1)}]-\left(\frac{3x-2}{x-1}\right)\left(TakingLCM\right)$

$=[\frac{2x^2-2x+5x-5}{(x-1)(x+1)}+\frac{x^2+1}{(x-1)(x+1)}]-\left(\frac{3x-2}{x-1}\right)$

$=\left[\frac{2x^2+3x-5+x^2+1}{(x-1)(x+1)}\right]-\left(\frac{3x-2}{x-1}\right)$

$=\frac{3x^2+3x-4}{(x-1)(x+1)}-\frac{3x-2}{x-1}$

$=\frac{3x^2+3x-4}{(x-1)(x+1)}-\frac{(3x-2)(x+1)}{(x-1)(x+1)}\left(TakingLCM\right)$

$=\frac{3x^2+3x-4}{(x-1)(x+1)}-\frac{3x^2+3x-2x-2}{(x-1)(x+1)}$

$=\frac{3x^2+3x-4}{(x-1)(x+1)}-\frac{3x^2+x-2}{(x-1)(x+1)}$

$=\frac{(3x^2+3x-4)-(3x^2+x-2)}{(x-1)(x+1)}=\frac{3x^2+3x-4-3x^2-x+2}{(x-1)(x+1)}$

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

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

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

Hence, $[\frac{2x+5}{x+1}+\frac{x^2+1}{x^2-1}]-\left(\frac{3x-2}{x-1}\right)=\frac{2}{x+1}$.