# 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 $a(x)={x}^{4}-2{x}^{3}+5$ 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 $a(x)={x}^{4}-2{x}^{3}+0{x}^{2}+0x+5$. 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

Title | polynomial long division |
---|---|

Canonical name | PolynomialLongDivision |

Date of creation | 2013-03-22 14:19:59 |

Last modified on | 2013-03-22 14:19:59 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 7 |

Author | rm50 (10146) |

Entry type | Definition |

Classification | msc 12D05 |

Related topic | LongDivision |