The Polynomial Factor Theorem is a theorem linking factors and zeros of a polynomial.[1] It is an application of the Polynomial Remainder Theorem. It states that a polynomial
has a factor
if and only if
. Here,
is also called the root of the polynomial.[2]
If
is a polynomial of a positive degree and if
so
is a factor of
.
According to the Polynomial Remainder Theorem, the remainder of the division of
by
is equal to
. As
, so the polynomial
is divisible by
∴
is a factor of
. [Proved]
Proposition : If
is a factor of the polynomial
then
Problem : Resolve the polynomial
into factors.
Solution : Here, the constant term of
is
and the set of the factors of
is
1
{±1, ±2}
Here, the leading coefficient of
is
and the set of the factors of
is
2
{±1, ±2, ±3, ±6, ±9, ±18}
Now consider
, where
1
2
When,
Therefore,
is a factor of
Now,
∴
Problem : Resolve the polynomial
into factors.
Solution : Considering only the terms of
and constant, we get
.
In the same way, considering only the terms of
and constant, we get
.
Combining factors of above (i) and (ii), the factors of the given polynomial can be found. But the constants
must remain same in both equations just like the coefficients of
and
.
∴
- ↑ [1] Byjus.com, Maths, Factor Theorem
- ↑ [2] Byjus.com, Maths, Roots of Polynomials