`x^4/(x - 1)^3` Write the partial fraction decomposition of the improper rational expression.

Expert Answers

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First make the substitution `x-1=y,` or `x=y+1.` Then the expression becomes `((y+1)^4)/(y^3).`

Because `(y+1)^4=y^4+4y^3+6y^2+4y+1,` the result is

`y+4+6/y+4/(y^2)+1/y^3.`

Recall `y=x-1` and obtain the answer,

`(x^4)/(x-1)^3=x+3+6/(x-1)+4/(x-1)^2+1/(x-1)^3.`

 

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