# grouping method for factoring polynomials

1. 1.

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2. 2.

Factorize the separately.

3. 3.

The whole polynomial may then possibly be written in form of a product.

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)   $x^{4}+4=\{x^{4}+4x^{2}+4\}-4x^{2}=(x^{2}+2)^{2}-(2x)^{2}=(x^{2}+2+2x)(x^{2}+2-% 2x)\\ =(x^{2}+2x+2)(x^{2}-2x+2)$

d)   $x^{4}+x^{2}+1=\{x^{4}+2x^{2}+1\}-x^{2}=(x^{2}+1)^{2}-x^{2}=(x^{2}+1+x)(x^{2}+1% -x)\\ =(x^{2}+x+1)(x^{2}-x+1)$

The trinomials $x^{2}\!+\!3x\!+\!1$, $x^{2}\!\pm\!2x\!+\!2$ and $x^{2}\!\pm\!x\!+\!1$ are irreducible polynomials  .

Title grouping method for factoring polynomials GroupingMethodForFactoringPolynomials 2013-03-22 15:06:49 2013-03-22 15:06:49 pahio (2872) pahio (2872) 12 pahio (2872) Algorithm msc 13P05 DifferenceOfSquares ExampleOfGcd ZeroRuleOfProduct grouping method