How do you write the partial fraction decomposition of the rational expression (x^3-x^2+1) / (x^4-x^3)x3−x2+1x4−x3?

1 Answer
mason m
Dec 24, 2015

1/(x-1)-1/x^2-1/x^31x−1−1x2−1x3

Explanation:

Factor the denominator.

(x^3-x^2+1)/(x^3(x-1))=A/x+B/x^2+C/x^3+D/(x-1)x3−x2+1x3(x−1)=Ax+Bx2+Cx3+Dx−1

Find a common denominator.

x^3-x^2+1=Ax^2(x-1)+Bx(x-1)+C(x-1)+Dx^3x3−x2+1=Ax2(x−1)+Bx(x−1)+C(x−1)+Dx3

x^3-x^2+1=Ax^3-Ax^2+Bx^2-Bx+Cx-C+Dx^3x3−x2+1=Ax3−Ax2+Bx2−Bx+Cx−C+Dx3

x^3-x^2+1=x^3(A+D)+x^2(-A+B)+x(-B+C)+1(-C)x3−x2+1=x3(A+D)+x2(−A+B)+x(−B+C)+1(−C)

Use this to write the system:

{(A+D=1),(-A+B=-1),(-B+C=0),(-C=1):}

Solve:

{(A=0),(B=-1),(C=-1),(D=1):}

Put this back in to the original expression:

(x^3-x^2+1)/(x^4-x^3)=1/(x-1)-1/x^2-1/x^3

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