Question

Write out the form of the partial fraction decomposition of the function (as in this example). Do not determine the numerical values of the coefficients. (a) x4 − 2x3 + x2 + 3x − 2 x2 − 2x + 1

Answer 1

$\dfrac{x^4-2x^3+x^2+3x-2}{x^2-2x+1}$

The degree of the numerator exceeds the degree of the denominator, so first you have to divide:

$x^2+\dfrac{3x-2}{x^2-2x+1}$

Now, $x^2-2x+1=(x-1)^2$, so the remainder term can be expanded to get

$\boxed{x^2+\dfrac a{x-1}+\dfrac b{(x-1)^2}}$