Factor theorem

In <a href="/facts/Algebra/nMdqczjo">algebra</a>, the factor theorem connects polynomial factors with <a href="/facts/Polynomial_root/mlK3dhoT">polynomial roots</a>. Specifically, if 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 is a polynomial, then 
 
 
 
 x
 −
 a
 
 
 {\displaystyle x-a}
 
 is a factor of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 if and only if 
 
 
 
 f
 (
 a
 )
 =
 0
 
 
 {\displaystyle f(a)=0}
 
 (that is, 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is a root of the polynomial). The theorem is a special case of the <a href="/facts/Polynomial_remainder_theorem/tTz57bPw">polynomial remainder theorem</a>.
The theorem results from basic properties of addition and multiplication. It follows that the theorem holds also when the coefficients and the element 
 
 
 
 a
 
 
 {\displaystyle a}
 
 belong to any <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>, and not just a <a href="/facts/Field_(algebra)/xAjAS4ko">field</a>. 
In particular, since multivariate polynomials can be viewed as univariate in one of their variables, the following generalization holds : If 
 
 
 
 f
 (
 
 X
 
 1
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 )
 
 
 {\displaystyle f(X_{1},\ldots ,X_{n})}
 
 and 
 
 
 
 g
 (
 
 X
 
 2
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 )
 
 
 {\displaystyle g(X_{2},\ldots ,X_{n})}
 
 are multivariate polynomials and 
 
 
 
 g
 
 
 {\displaystyle g}
 
 is independent of 
 
 
 
 
 X
 
 1
 
 
 
 
 {\displaystyle X_{1}}
 
, then 
 
 
 
 
 X
 
 1
 
 
 −
 g
 (
 
 X
 
 2
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 )
 
 
 {\displaystyle X_{1}-g(X_{2},\ldots ,X_{n})}
 
 is a factor of 
 
 
 
 f
 (
 
 X
 
 1
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 )
 
 
 {\displaystyle f(X_{1},\ldots ,X_{n})}
 
 if and only if 
 
 
 
 f
 (
 g
 (
 
 X
 
 2
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 )
 ,
 
 X
 
 2
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 )
 
 
 {\displaystyle f(g(X_{2},\ldots ,X_{n}),X_{2},\ldots ,X_{n})}
 
 is the zero polynomial.

Factor theorem open-in-new

Factor theorem