How do you write the partial fraction decomposition of the rational expression #(x^2 − x + 1 )/( x^3 − x^2 + x − 1)#?
1 Answer
Aug 7, 2016
Explanation:
#(x^2-x+1)/(x^3-x^2+x-1)#
#=(x^2-x+1)/((x-1)(x^2+1))#
#=A/(x-1)+(Bx+C)/(x^2+1)#
#=(A(x^2+1)+(Bx+C)(x-1))/(x^3-x^2+x-1)#
#=((A+B)x^2+(C-B)x+(A-C))/(x^3-x^2+x-1)#
Equating coefficients, we get:
#{ (A+B=1), (C-B=-1), (A-C=1) :}#
Hence:
#{ (A=1/2), (B=1/2), (C=-1/2) :}#
So:
#(x^2-x+1)/(x^3-x^2+x-1)=1/(2(x-1))+(x-1)/(2(x^2+1))#
