Write the partial fraction decomposition of the rational exp

Write the partial fraction decomposition of the rational expression. Check your result algebraically.

2x^2 8x + 12 / (x + 3)(x2 2x + 3)

Solution

(2x^2 8x + 12) / (x + 3)(x2 2x + 3)

This cannot be factorised further:

If i take the expression : -2x^2 -8x +12/(x^2 -2x +3)

: -2x^2 -8x +12/(x^2 -2x +3) = -2x^2 -8x +12/(x^2 -2x +3)(x+3)

So, 12/(x^2 -2x +3)(x+3) = (a+bx)/(x^2 -2x +3) + c/(x+3)

12 = (x+bx)(x+3) + x(x^2 -2x +3)

Multiplying the terms on RHS and equating with coefficients on LHS

3a+2 =12 ;

a+3b -4/3 =0

a +2/3 =0

On solving we get c = 2/3 ; a = 10/3 ; b=-2/3

So, 12/(x^2 -2x +3)(x+3) = (a+bx)/(x^2 -2x +3) + c/(x+3)

= (10/3 -2x/3)/(x^2 -2x +3) +2/3(x+3)

= 10-2x/3(x^2 -2x +3) +2/3(x+3)

Factorised form : -2x^2 -8x +12/(x^2 -2x +3) = -2x^2 - 8x + (10-2x)/3(x^2 -2x +3) +2/3(x+3)

if we again add the terms of RHs we would get back to LHS