# 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 *x*^{3} + 3*x*^{2} - 16*x* + 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 *x*^{4} - 2*x*^{3} - 4*x*^{2} - 2*x* + 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 *x*^{4} + *x*^{3} - 5*x*^{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 (*x*^{2} + 0*x* + 1).

So (given) = (*x* - 2)(*x* - (-3))(*x*^{2} + 1).

-(-3) = +3

So (given) = (*x* - 2)(*x* + 3)(*x*^{2} + 1).