Difference between revisions of "Factor Theorem"
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==Statement== | ==Statement== | ||
− | The '''Factor Theorem''' says that if <math>P(x)</math> is a [[polynomial]], then <math>x-a</math> is a [[factor]] of <math>P(x)</math> if <math>P(a)=0</math>. | + | The '''Factor Theorem''' says that if <math>P(x)</math> is a [[polynomial]], then <math>x-a</math> is a [[factor]] of <math>P(x)</math> if and only if <math>P(a)=0</math>. |
==Proof== | ==Proof== |
Revision as of 21:48, 13 January 2024
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
Olympaid
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