How do you write the partial fraction decomposition of the rational expression (x^2 − x + 1 )/( x^3 − x^2 + x − 1)?

1 Answer
George C.
Aug 7, 2016

(x^2-x+1)/(x^3-x^2+x-1)=1/(2(x-1))+(x-1)/(2(x^2+1))

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))

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