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

`(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+(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+\frac a{x-1}+\frac b{(x-1)^2}}`