Factor Theorem
How to use the factor theorem to factor polynomials: theorem, examples and their solutions.
Theorem
If f(a) = 0,
then (x - a) is the factor of f(x).
This is the factor theorem.
Extending the Factor Theorem
So if f(a) = 0 and f(b) = 0,
then (x - a) and (x - b) are the factors of f(x).
This means
by finding the zeros of f(x),
you can factor f(x).
Example 1: Factor x3 + 3x2 - 16x + 12
Use synthetic division
to find the zeros of the given polynomial.
Write the coefficients of the terms:
1 3 -16 12.
Write the L shape form like this.
Pick a number
that seems to be the zero:
that makes the remainder 0.
1 seems to be a good start.
So write 1
next to the form.
Synthetic division
↓: 1
↗: 1⋅1 = 1
↓: 3 + 1 = 4
↗: 4⋅1 = 4
↓: -16 + 4 = -12
↗: -12⋅1 = -12
↓: 12 - 12 = 0
The remainder is 0.
So 1 is the zero of the polynomial.
Do synthetic division again.
Again, pick a number
that seems to be the zero:
that makes the remainder 0.
2 seems to be the zero.
So write 2
next to the form.
↓: 1
↗: 1⋅2 = 2
↓: 4 + 2 = 6
↗: 6⋅2 = 12
↓: -12 + 12 = 0
The remainder is 0.
So 2 is another zero of the polynomial.
1 and 2 are the zeros.
(blue)
So (x - 1) and (x - 2)
are the factors of the polynomial.
[1 6] means the quotient (x + 6).
So (given) = (x - 1)(x - 2)(x + 6).
Example 2: Factor x4 - 2x3 - 4x2 - 2x + 3
Use synthetic division
to find the zeros of the given polynomial.
Write the coefficients of the terms:
1 -2 -4 2 3.
Write the L shape form like this.
Pick a number
that seems to make the remainder 0.
1 seems to be the zero.
So write 1
next to the form.
↓: 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 1 is the zero of the polynomial.
Do synthetic division again.
Again, pick a number
that seems to make the remainder 0.
-1 seems to be the zero.
So write -1
next to the form.
↓: 1
↗: 1⋅(-1) = -1
↓: -1 - 1 = -2
↗: -2⋅(-1) = 2
↓: -5 + 2 = -3
↗: -3⋅(-1) = 3
↓: -3 + 3 = 0
The remainder is 0.
So -1 is the zero of the polynomial.
Do synthetic division again.
Pick a number
that seems to make the remainder 0.
Again, -1 seems to be the zero.
So write -1
next to the form.
(It's okay to pick the same number again.)
↓: 1
↗: 1⋅(-1) = -1
↓: -2 - 1 = -3
↗: -3⋅(-1) = 3
↓: -3 + 3 = 0
The remainder is 0.
So -1 is the zero of the polynomial.
1, -1, and -1 are the zeros.
(blue)
So (x - 1), (x - (-1)), and (x - (-1))
are the factors of the polynomial.
[1 -3] means the quotient (x - 3).
So (given) = (x - 1)(x - (-1))2(x - 3).
-(-1) = +1
So (given) = (x - 1)(x + 1)2(x - 3).
Example 3: Factor x4 + x3 - 5x2 + x - 6
Use synthetic division
to find the zeros of the given polynomial.
Write the coefficients of the terms:
1 1 -5 1 -6.
Write the L shape form like this.
Pick a number
that seems to make the remainder 0.
2 seems to be the zero.
So write 2
next to the form.
↓: 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 2 is the zero of the polynomial.
Do synthetic division again.
Pick a number
that seems to make the remainder 0.
-3 seems to be the zero.
So write -3
next to the form.
↓: 1
↗: 1⋅(-3) = -3
↓: 3 - 3 = 0
↗: 0⋅(-3) = 0
↓: 1 + 0 = 1
↗: 1⋅(-3) = -3
↓: 3 - 3 = 0
The remainder is 0.
So -3 is the zero of the polynomial.
2 and -3 are the zeros.
(blue)
So (x - 2) and (x - (-3))
are the factors of the polynomial.
[1 0 1] means the quotient (x2 + 0x + 1).
So (given) = (x - 2)(x - (-3))(x2 + 1).
-(-3) = +3
So (given) = (x - 2)(x + 3)(x2 + 1).
