How do you write the partial fraction decomposition of the rational expression #x^3/(x^2 + 4x + 3) #?
1 Answer
Aug 2, 2016
#x^3/(x^2+4x+3)=x-4-1/(2(x+1))+27/(2(x+3))#
Explanation:
#x^3/(x^2+4x+3)#
#=((x^3+4x^2+3x)-(4x^2+16x+12)+(13x+12))/(x^2+4x+3)#
#=x-4+(13x+12)/(x^2+4x+3)#
#=x-4+(13x+12)/((x+1)(x+3))#
#=x-4+A/(x+1)+B/(x+3)#
Using Heaviside's cover-up method we find:
#A = (13(-1)+12)/((-1)+3) = -1/2#
#B = (13(-3)+12)/((-3)+1) = 27/2#
So:
#x^3/(x^2+4x+3)=x-4-1/(2(x+1))+27/(2(x+3))#
