Factor Theorem

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).

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).

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

Factor the given polynomial. x^3 + 3x^2 - 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

Factor the given polynomial. x^4 - 2x^3 - 4x^2 - 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

Factor the given polynomial. x^4 + x^3 - 5x^2 + 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).