In algebra, the factor theorem connects polynomial factors with polynomial roots.
The theorem is a special case of the polynomial remainder theorem.
[1][2] The theorem results from basic properties of addition and multiplication.
It follows that the theorem holds also when the coefficients and the element
belong to any commutative ring, and not just a field.
In particular, since multivariate polynomials can be viewed as univariate in one of their variables, the following generalization holds : If
Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent.
The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find.
Abstractly, the method is as follows:[3] Continuing the process until the polynomial
Out of these, the quadratic factor can be further factored using the quadratic formula, which gives as roots of the quadratic
Thus the three irreducible factors of the original polynomial are
Several proofs of the theorem are presented here.
So, only the converse will be proved in the following.
This proof begins by verifying the statement for
To that end, write
This case is now proven.
What remains is to prove the theorem for general
To that end, observe that
belong to any commutative ring (the same one) then the identity
This is shown by multiplying out the brackets.
is any commutative ring.
for a sequence of coefficients
Observe that each summand has
as a factor by the factorisation of expressions of the form
The theorem may be proved using Euclidean division of polynomials: Perform a Euclidean division of
Finally, observe that
The Euclidean division above is possible in every commutative ring since
is a monic polynomial, and, therefore, the polynomial long division algorithm does not involve any division of coefficients.
It is also a corollary of the polynomial remainder theorem, but conversely can be used to show it.
When the polynomials are multivariate but the coefficients form an algebraically closed field, the Nullstellensatz is a significant and deep generalisation.