Question

Factorize it partially
$\frac{1+2x^2}{x^2(1+x^2)}$

• A. $\frac{1+x}{x^2}+\frac{1}{1+x^2}$
• B. $\frac{1}{x^2}+\frac{1}{1+x^2}$
• C. $\frac{1}{x^2}-\frac{x}{1+x^2}$
• D. $\frac{1+x}{x^2}-\frac{x}{1+x^2}$

Answer 1

The correct option is B $\frac{1}{x^2}+\frac{1}{1+x^2}$

Let $\frac{1+2x^2}{x^2(1+x^2)}=\frac{A}{x}+\frac{B}{x^2}+\frac{Cx+D}{1+x^2}$ .....(1)

Multiply $x^2(1+x^2)$ on both the sides, we get

$1+2x^2=Ax(1+x^2)+B(1+x^2)+(Cx+D)\left(x^2\right)$

Solving this, we get

$A+C=0,B+D=2,A=0,B=1$

On solving, we get $C=0,D=1$

Substituting the values of A,B,C and D in equation (1), we get

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