# Polynomial long division(alternative methods)

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?

You are correct in dividing $$g(x)$$ by $$h(x)$$ when the degree of $$g(x)$$ is greater than or equal the degree of $$h(x)$$
We need to find partial fractions for $$r(x)/{h(x)}$$ in order to proceed with integration.