Editing Factorization (section)

===Using the factor theorem===
{{Main|Factor theorem}}

The factor theorem states that, if {{mvar|r}} is a [[zero of a function|root]] of a [[polynomial]]
:<math>P(x)=a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n,</math>
meaning {{math|1=''P''(''r'') = 0}}, then there is a factorization 
:<math>P(x)=(x-r)Q(x),</math>
where 
:<math>Q(x)=b_0x^{n-1}+\cdots+b_{n-2}x+b_{n-1},</math>
with <math>a_0=b_0</math>. Then [[polynomial long division]] or [[synthetic division]] give:
:<math>b_i=a_0r^i +\cdots+a_{i-1}r+a_i \ \text{ for }\ i = 1,\ldots,n{-}1.</math>
This may be useful when one knows or can guess a root of the polynomial.

For example, for <math>P(x) = x^3 - 3x + 2,</math> one may easily see that the sum of its coefficients is 0, so {{math|1=''r'' = 1}} is a root. As {{math|1=''r'' + 0 = 1}}, and <math>r^2 +0r-3=-2,</math> one has 
:<math>x^3 - 3x + 2 = (x - 1)(x^2 + x - 2).</math>