polynomial long divisionPolynomialLongDivision2004-04-25 03:17:572007-12-30 19:35:43Definitionroot% this is the default PlanetMath preamble. as your knowledge
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% define commands hereGiven two polynomials $a(x)$ and $b(x)$ \emph{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 $\sum_{i=0}^{n} c_ix^i$ 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
\includegraphics{pd.eps}
In the next step we se that $1/1=1$ and we multiply 1 3 -2 with 1 and then subtract the result.
\includegraphics{pd1.eps}
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
\includegraphics{pd2.eps}
as a final result we get
\includegraphics{pd3.eps}
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
\includegraphics{pd4.eps}