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[parent] polynomial long division (Definition)

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_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}



"polynomial long division" is owned by rm50. [ full author list (2) | owner history (1) ]
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See Also: long division


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Cross-references: entire, translates, number, coefficients, integer, similar, division, polynomials
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This is version 4 of polynomial long division, born on 2004-04-25, modified 2007-12-30.
Object id is 5803, canonical name is PolynomialLongDivision.
Accessed 8559 times total.

Classification:
AMS MSC: 12D05 (Field theory and polynomials :: Real and complex fields :: Polynomials: factorization)
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