5 4 Repeated Quadratic Factors – Class 11TH Math

Case IV: Repeated Irreducible Quadratic Factors

If $Q(x)$ contains $(ax^2+bx+c)^n$ with $n \ge 2$ , then

\dfrac{P(x)}{Q(x)} = \dfrac{A_1 x + B_1}{ax^2+bx+c} + \dfrac{A_2 x + B_2}{(ax^2+bx+c)^2} + \cdots + \dfrac{A_n x + B_n}{(ax^2+bx+c)^n}

Resolve $\dfrac{4x^2}{(x^2+1)^2(x-1)}$ into partial fractions.

Solution

\dfrac{4x^2}{(x^2+1)^2(x-1)} = \dfrac{Ax+B}{x^2+1} + \dfrac{Cx+D}{(x^2+1)^2} + \dfrac{E}{x-1}

4x^2 = (Ax+B)(x^2+1)(x-1) + (Cx+D)(x-1) + E(x^2+1)^2

$x = 1$ $\Rightarrow$ $E = 1$ . Equating coefficients: $A = -1$ , $B = -1$ , $C = 2$ , $D = 2$ .

Hence

\dfrac{-x-1}{x^2+1} + \dfrac{2x+2}{(x^2+1)^2} + \dfrac{1}{x-1}