Question

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

Answer 1

Factorizing the denominator gives

$\dfrac{x^4}{(x^3+x)(x^2-x+7)} = \dfrac{x^4}{x(x^2+1)(x^2-x+7)} = \dfrac{x^3}{(x^2+1)(x^2-x+7)}$

Then the partial fraction decomposition would take the form

$\dfrac{x^4}{(x^3+x)(x^2-x+7)} = \dfrac{a_0x + b_0}{x^2+1} + \dfrac{a_1x + b_1}{x^2-x+7}$