Factor Theorem
How to use the factor theorem to factor a polynomial: theorem, formula, 3 examples, and their solutions.
Theorem
Theorem
If f(a) = 0,
then f(x) has the factor (x - a).
This is the factor theorem.
Then if f(a) = 0, f(b) = 0, ...,
then f(x) has the factors (x - a), (x - b), ... .
So, by finding the zeros,
you can factor f(x).
Formula
Formula
Recall that
f(a) is the remainder of the synthetic division
of f(x)/(x - a).
Synthetic Substitution
So, if the remainder is 0, f(a) = 0,
then (x - a) is the factor of f(x).
By repeating this,
you can find the factors of f(x)
and factor f(x).
Example 1
Example
Solution
Do the synthetic division.
f(x) = x3 + 3x2 - 16x + 12
Write the coefficients of the terms
in descending order:
1 3 -16 12.
Draw an L shape form like this.
Pick a number
that seems to make the remainder, f(a), 0.
1 seems to be good.
So write 1
on the left side of the form.
Tip: If the sum of the coefficients is 0,
then 1 is the right number.
↓: 1 = 1
↗: 1⋅1 = 1
↓: 3 + 1 = 4
↗: 4⋅1 = 4
↓: -16 + 4 = -12
↗: -12⋅1 = -12
↓: 12 - 12 = 0
The remainder is 0.
So f(1) = 0.
So 1 is the right number.
If the remainder is not 0,
then you picked the wrong number.
Then pick another number.
This is a process of trial and error.
So it's natural to choose the wrong numbers.
1 4 -12 means
x2 + 4x - 12.
This seems to be factorable.
So do the synthetic division again.
Draw an L shape form like this.
Pick a number
that seems to make the remainder 0.
2 seems to be good.
So write 2
on the left side of the form.
↓: 1 = 1
↗: 1⋅2 = 2
↓: 4 + 2 = 6
↗: 6⋅2 = 12
↓: 12 - 12 = 0
The remainder is 0.
So f(2) = 0.
So 2 is the right number.
1 6 means
x + 6.
(x + 6) is a binomial factor.
So stop finding the factors.
And write the factors
from this.
For the number 1,
the remainder, f(1), is 0.
So write the factor (x - 1).
For the number 2,
the remainder, f(2), is 0.
So write the factor (x - 2).
1 6 means
x + 6.
So write (x + 6).
This is the quotient.
So (given) = (x - 1)(x - 2)(x + 6).
So
(x - 1)(x - 2)(x + 6)
is the answer.
Example 2
Example
Solution
Do the synthetic division.
f(x) = x4 - 2x3 - 4x2 - 2x + 3
Write the coefficients of the terms
in descending order:
1 -2 -4 -2 3.
Draw an L shape form like this.
Pick a number
that seems to make the remainder 0.
1 seems to be good.
So write 1
on the left side of the form.
↓: 1 = 1
↗: 1⋅1 = 1
↓: -2 + 1 = -1
↗: -1⋅1 = -1
↓: -4 - 1 = -5
↗: -5⋅1 = -5
↓: 2 - 5 = -3
↗: -3⋅1 = -3
↓: 3 - 3 = 0
The remainder is 0.
So f(1) = 0.
So 1 is the right number.
1 -1 -5 3 means
x3 - x2 - 5x + 3.
This seems to be factorable.
So do the synthetic division again.
Draw an L shape form like this.
Pick a number
that seems to make the remainder 0.
-1 seems to be good.
So write -1
on the left side of the form.
↓: 1 = 1
↗: 1⋅(-1) = -1
↓: -1 - 1 = -2
↗: -2⋅(-1) = 2
↓: -5 + 2 = -3
↗: -3⋅(-1) = 3
↓: -3 + 3 = 0
The remainder is 0.
So f(-1) = 0.
So -1 is the right number.
1 -2 -3 means
x2 - 2x - 3.
This seems to be factorable.
So do the synthetic division again.
Draw an L shape form like this.
Pick a number
that seems to make the remainder 0.
-1 seems to be good.
So write -1 again
on the left side of the form.
↓: 1 = 1
↗: 1⋅(-1) = -1
↓: -2 - 1 = -3
↗: -3⋅(-1) = 3
↓: -3 + 3 = 0
The remainder is 0.
So f(-1) = 0.
So -1 is the right number.
1 -3 means
x - 3.
(x - 3) is a binomial factor.
So stop finding the factors.
And write the factors
from this.
For the number 1,
the remainder, f(1), is 0.
So write the factor (x - 1).
For two numbers -1,
the remainders are both 0.
So write the factor (x + 1)2.
1 -3 means
x - 3.
So write (x - 3).
This is the quotient.
So (given) = (x - 1)(x + 1)2(x - 3).
So
(x - 1)(x + 1)2(x - 3)
is the answer.
Example 3
Example
Solution
Do the synthetic division.
f(x) = x4 + x3 - 5x2 + x + 6
Write the coefficients of the terms
in descending order:
1 1 -5 1 -6.
Draw an L shape form like this.
Pick a number
that seems to make the remainder 0.
2 seems to be good.
So write 2
on the left side of the form.
↓: 1 = 1
↗: 1⋅2 = 2
↓: 1 + 2 = 3
↗: 3⋅2 = 6
↓: -5 + 6 = 1
↗: 1⋅2 = 2
↓: 1 + 2 = 3
↗: 3⋅2 = 6
↓: -6 + 6 = 0
The remainder is 0.
So f(2) = 0.
So 2 is the right number.
1 3 1 3 means
x3 + 3x2 + x + 3.
This seems to be factorable.
So do the synthetic division again.
Draw an L shape form like this.
Pick a number
that seems to make the remainder 0.
-3 seems to be good.
So write -3
on the left side of the form.
↓: 1 = 1
↗: 1⋅(-3) = -3
↓: 3 - 3 = 0
↗: 0⋅(-3) = 0
↓: 1 + 0 = 1
↗: 1⋅(-3) = -3
↓: 3 - 3 = 0
The remainder is 0.
So f(-3) = 0.
So -3 is the right number.
1 0 1 means
x2 + 1.
x2 + 1 is always (+).
There's no root for x2 + 1 = 0
So this is not factorable.
So stop the synthetic division.
And write the factors
from this.
For the number 2,
the remainder, f(2), is 0.
So write the factor (x - 2).
For the number -3,
the remainder, f(-3), is 0.
So write the factor (x + 3).
1 0 1 means
x2 + 1.
So write (x2 + 1).
This is the quotient.
So (given) = (x - 2)(x + 3)(x2 + 1).
So
(x - 2)(x + 3)(x2 + 1)
is the answer.