Polynomial long division(alternative methods)

Question

I used the book Mathematical methods for physics and engineering. In the algebra section, it uses a method for dividing polynomials I have never seen for decomposing $\frac{g(x)}{h(x)}$ into $s(x)+\frac{r(x)}{h(x)}$ (basically what long division is used for). Instead of using the traditional algorithm, it does the following manipulation: $$\frac{g(x)}{h(x)} = s(x)+\frac{r(x)}{h(x)}$$ therefore, $$g(x) = h(x)s(x)+r(x)$$ Putting it explicitly; if $g(x)$ has degree $m$ and $h(x)$ has degree $n$: $$g(x) = (s_{m-n}x^{m-n}+s_{m-n-1}x^{m-n-1}+...+s_0)h(x)+(r_{n-1}x^{n-1}+r_{n-2}x^{n-2}+...+r_0)$$ Then, equate the coefficients of $x^m,x^{m-1},...x^{(1)},x^0$ (constant coefficient). So with that, the polynomials $r(x)$ and $s(x)$ are now known. Which do you think is the best method to use for polynomial division?

Answer 1

You are correct in dividing $g(x)$ by $h(x)$ when the degree of $g(x)$ is greater than or equal the degree of $h(x)$

The actual partial fraction decomposition starts after the dividing is done.

We need to find partial fractions for $r(x)/{h(x)}$ in order to proceed with integration.