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 $q(x)=1\ -5\ 17$, 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

Title polynomial long division PolynomialLongDivision 2013-03-22 14:19:59 2013-03-22 14:19:59 rm50 (10146) rm50 (10146) 7 rm50 (10146) Definition msc 12D05 LongDivision