polynomial long division

Given two polynomials $a(x)$ and $b(x)$ polynomial (long) division is a method for calculating $a(x)/b(x)$ that is, finding the polynomials $q(x)$ and $r(x)$ such that $a(x)=b(x)q(x)+r(x)$.

Here is an example to show the method.Let $a(x)=x^4-2x^3+5$ and $b(x)=x^2+3x-2$.The method looks very similar to integer division since a polynomial ${\sum }_{i=0}^n{c}_i{x}^i$ is somewhat similar to an integer ${\sum }_{i=0}^n{c}_i{10}^i$

In the initial setting we only write the coefficients, notice that $a(x)=x^4-2x^3+0x^2+0x+5$. It will then be

In the next step we se that $1/1=1$ and we multiply 1 3 -2 with 1 and then subtract the result.

Then we move down the next number, in this case a zero, and $-5/1=-5$ so we get -5, and multiply by -5 and subtract

as a final result we get

The result is $q(x)=1-\mathrm{5\hspace{0.25em}17}$, which translates to $q(x)=x^2-5x+17$ and $r(x)=-61x+39$.

It is also possible to write the entire polynomial, that is, writing all the $x^i$’s. Like this