How do you write the partial fraction decomposition of the rational expression 1/((x^4)-6(x^3))?

1 Answer
Dec 14, 2015

1/(216(x-6))-1/(216x)-1/(36x^2)-1/(6x^3)

Explanation:

Factor the denominator.

1/(x^3(x-6))=A/x+B/x^2+C/x^3+D/(x-6)

1=A(x^2)(x-6)+B(x)(x-6)+C(x-6)+D(x^3)

1=Ax^3-6Ax^2+Bx^2-6Bx+Cx-6C+Dx^3

1=x^3(A+D)+x^2(-6A+B)+x(-6B+C)+1(-6C)

Use this to write the following system:

{(A+D=0),(-6A+B=0),(-6B+C=0),(-6C=1):}

Solve to see that:

{(A=-1/216),(B=-1/36),(C=-1/6),(D=1/216):}

The fraction decomposes into:

1/(216(x-6))-1/(216x)-1/(36x^2)-1/(6x^3)