Write out the form of the partial fraction decomposition of the fraction. Do not determine the numerical values of the coefficients. (x^4)/((x^3+x)(x^2-x+7))

Question

Write out the form of the partial fraction decomposition of the fraction. Do not determine the numerical values of the coefficients.


`(x^4)/((x^3+x)(x^2-x+7))`

Answer 1

Factorizing the denominator gives

`(x^4)/((x^3+x)(x^2-x+7)) = (x^4)/(x(x^2+1)(x^2-x+7)) = (x^3)/((x^2+1)(x^2-x+7))`

Then the partial fraction decomposition would take the form

`(x^4)/((x^3+x)(x^2-x+7)) = (a_0x + b_0)/(x^2+1) + (a_1x + b_1)/(x^2-x+7)`