Editing Factorization (section)

===Using relations between roots===
It may occur that one knows some relationship between the roots of a polynomial and its coefficients. Using this knowledge may help factoring the polynomial and finding its roots. [[Galois theory]] is based on a systematic study of the relations between roots and coefficients, that include [[Vieta's formulas]].

Here, we consider the simpler case where two roots <math>x_1</math>
and <math>x_2</math> of a polynomial <math>P(x)</math> satisfy the relation
:<math>x_2=Q(x_1),</math>
where {{mvar|Q}} is a polynomial.

This implies that <math>x_1</math> is a common root of <math>P(Q(x))</math> and <math>P(x).</math> It is therefore a root of the [[Polynomial greatest common divisor|greatest common divisor]] of these two polynomials. It follows that this greatest common divisor is a non constant factor of <math>P(x).</math> [[Euclidean algorithm for polynomials]] allows computing this greatest common factor.

For example,<ref>{{harvnb|Burnside|Panton|1960|p=38}}</ref> if one know or guess that:
<math>P(x)=x^3 -5x^2 -16x +80</math> 
has two roots that sum to zero, one may apply Euclidean algorithm to <math>P(x)</math> and <math>P(-x).</math> The first division step consists in adding <math>P(x)</math> to <math>P(-x),</math> giving the [[remainder]] of 
:<math>-10(x^2-16).</math>
Then, dividing <math>P(x)</math> by <math>x^2-16</math> gives zero as a new remainder, and {{math|''x'' − 5}} as a quotient, leading to the complete factorization
:<math>x^3 - 5x^2 - 16x + 80 = (x -5)(x-4)(x+4).</math>