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| Title of object: |
grouping method for factorizing polynomials |
| Canonical Name: |
GroupingMethodForFactorizingPolynomials |
| Type: |
Algorithm |
| Created on: |
2005-03-06 17:06:38 |
| Modified on: |
2005-03-06 17:06:38 |
| Classification: |
msc:13P05 |
Preamble:
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Content:
Factorizing a given polynomial may in certain special cases \PMlinkescapetext{succeed} by using the following {\em grouping method}:
\begin{enumerate}
\item \PMlinkescapetext{Group the terms of the polynomial in two suitable groups}.
\item Factorize the \PMlinkescapetext{groups} separately.
\item The whole polynomial may then possibly be written in form of a product.
\end{enumerate}
\textbf{Examples}
a) \,\,$x^3-x^2-x+1 = \{x^3-x^2\}+\{-x+1\} = x^2(x-1)-1(x-1) = (x-1)(x^2-1)\\ = (x-1)^2(x+1)$
b) \,\,$x^4+3x^3-3x-1 = \{x^4-1\}+\{3x^3-3x\} = (x^2+1)(x^2-1)+3x(x^2-1)\\ =
(x^2-1)(x^2+1+3x) = (x-1)(x+1)(x^2+3x+1)$
c) \,\,$x^4+4 = \{x^4+4x^2+4\}-4x^2 = (x^2+2)^2-(2x)^2 = (x^2+2+2x)(x^2+2-2x)\\ = (x^2+2x+2)(x^2-2x+2)$
The trinomials $x^2+3x+1$ and $x^2\pm 2x+2$ are irreducible polynomials. |
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