Factor theorem

Factor theorem
In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if is a polynomial, then is a factor of if and only if (that is, is a root of the polynomial). The theorem is a special case of the polynomial remainder theorem.
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 and are multivariate polynomials and is independent of , then is a factor of if and only if is the zero polynomial.
Factorization of polynomials
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:
Deduce the candidate of zero of the polynomial from its leading coefficient and constant term . (See Rational Root Theorem.)
Use the factor theorem to conclude that is a factor of .
Compute the polynomial , for example using polynomial long division or synthetic division.
Conclude that any root of is a root of . Since the polynomial degree of is one less than that of , it is "simpler" to find the remaining zeros by studying .
Continuing the process until the polynomial is factored completely, which all its factors is irreducible on or .
Example
Find the factors of
Solution: Let be the above polynomial
Constant term = 2
Coefficient of
All possible factors of 2 are and . Substituting , we get:
So i.e, is a factor of . On dividing by , we get
Quotient =
Hence,
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 and
Proof
Several proofs of the theorem are presented here.
If is a factor of it is immediate that So, only the converse will be proved in the following.
Proof 1
This argument begins by verifying the theorem for . That is, it aims to show that for any polynomial for which it is true that for some polynomial . To that end, write explicitly as . Now observe that , so . Thus, . This case is now proven.
What remains is to prove the theorem for general by reducing to the case. To that end, observe that is a polynomial with a root at . By what has been shown above, it follows that for some polynomial . Finally, .
Proof 2
First, observe that whenever and belong to any commutative ring (the same one) then the identity is true. This is shown by multiplying out the brackets.
Let where is any commutative ring. Write for a sequence of coefficients . Assume for some . Observe then that . Observe that each summand has as a factor by the factorisation of expressions of the form that was discussed above. Thus, conclude that is a factor of .
Proof 3
The theorem may be proved using Euclidean division of polynomials: Perform a Euclidean division of by to obtain where . Since , it follows that is constant. Finally, observe that . So .
The Euclidean division above is possible in every commutative ring since is a monic polynomial, and, therefore, the polynomial long division algorithm does not involves any division of coefficients.
Corollary of other theorems
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.
References
Theorems about polynomials