Polynomial zeros related to linear factors
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.[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
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:[3]
- 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
![{\displaystyle x^{3}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1990361bf76c23e7feaacd57da472eb626ad5760)
All possible factors of 2 are
and
. Substituting
, we get:
![{\displaystyle (-1)^{3}+7(-1)^{2}+8(-1)+2=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d290daa0b7a7d76e77bd71be83f464fccd28f27)
So,
, i.e,
is a factor of
. On dividing
by
, we get
- Quotient =
![{\displaystyle x^{2}+6x+2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c228f90c547e1dee163fbf0a5f45a7bb4679661)
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.