Question

Split into partial fractions
$\frac{3+2x-x^2}{(1+x){(1-4x)}^2}$

Answer 1

To find the general term of the following expressions when expanded in ascending powers of x

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

Here , we have

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

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

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

The Expression can be written as

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

On Solving for $A,B$ we have

$A=\frac{1}{4},B=\frac{11}{4}$

Therefore , The Above Expression as

$=\frac{1}{4(1-4x)}+\frac{11}{4(1-4x{)}^2}$