| Version 3 |
Version 2 |
| Given 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 $a(x)=b(x)q(x)+r(x)$. |
Given 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 $b(x)=x^2+3x-2$. |
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 $\sum_{i=0}^{n} c_i 10^i$ |
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 |
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} |
\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. |
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} |
\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 |
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} |
\includegraphics{pd2.eps} |
|
|
| as a final result we get |
as a final result we get |
|
|
|
|
| \includegraphics{pd3.eps} |
\includegraphics{pd3.eps} |
|
|
|
|
|
The result is $q(x)=1\ -5\ 17$, which translates to $q(x)=x^2-5x+17$ and $r(x)=-61x+39$.
|
The result is $q(x)=1 -5 7$, which translates to $q(x)=x^2-5x+7$ and $r(x)=-61x+39$.
|
|
|
| It is also possible to write the entire polynomial, that is, writing all the $x^i$'s. Like this |
It is also possible to write the entire polynomial, that is, writing all the $x^i$'s. Like this |
|
|
| \includegraphics{pd4.eps} |
\includegraphics{pd4.eps} |
