Factor Theorem
In algebra, the Factor theorem is a theorem regarding the relationships between the factors of a polynomial and its roots.
One of it's most important applications is if you are given that a polynomial have certain roots, you will know certain linear factors of the polynomial. Thus, you can test if a linear factor is a factor of a polynomial without using polynomial division and instead plugging in numbers. Conversely, you can determine whether a number in the form (
is constant,
is polynomial) is
using polynomial division rather than plugging in large values.
Statement
The Factor Theorem says that if is a polynomial, then
is a factor of
if and only if
.
Proof
If is a factor of
, then
, where
is a polynomial with
. Then
.
Now suppose that .
Apply Remainder Theorem to get , where
is a polynomial with
and
is the remainder polynomial such that
. This means that
can be at most a constant polynomial.
Substitute and get
. Since
is a constant polynomial,
for all
.
Therefore, , which shows that
is a factor of
.
Problems
Here are some problems that can be solved using the Factor Theorem:
Introductory
Intermediate
Suppose is a
-degrees polynomial. The Fundamental Theorem of Algebra tells us that there are
roots, say
. Suppose all integers
ranging from
to
satisfies
. Also, suppose that
for an integer . If
is the minimum possible positive integral value of
.
Find the number of factors of the prime in
. (Source: I made it. Solution here)
Olympaid
If denotes a polynomial of degree
such that
for
, determine
.
(Source: 1975 USAMO Problem 3)
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