Factor Theorem

Factor Theorem

How to use the factor theorem to factor polynomials: theorem, proof, 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).

Proof

Factor Theorem: Proof

f(x) = (x - a)⋅(quotient) + f(a)

Remainder theorem

If f(a) = 0,
then f(x) = (x - a)⋅(quotient).

So if f(a) = 0,
then (x - a) is the factor of f(x).

How to Use

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 you can factor a polynomial
by finding the zeros of the function.

Example 1

Factor the given polynomial. x^3 + 3x^2 - 16x + 12

To factor the given polynomial,
find the zeros of the function.

The zeros are the numbers
that make the remainder of the synthetic division 0.

Use the synthetic division.
(divisor's zero: 1)

The remainder is 0.
Then f(1) = 0.

Synthetic substitution

So (x - 1) is the factor of f(x).

To factor further,
use the synthetic division.
(divisor's zero: 2)

The remainder is 0.
Then f(2) = 0.

So (x - 2) is the factor of f(x).

1, 6 means (x + 6). (brown)

(x + 6) is the factor that can't be factored further.

The previously found zeros are 1 and 2.

So f(x) = (x - 1)(x - 2)(x + 6).

Example 2

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

Use the synthetic division.
(divisor's zero: 1)

The remainder is 0.
Then f(1) = 0.

Synthetic substitution

So (x - 1) is the factor of f(x).

Use the synthetic division.
(divisor's zero: -1)

The remainder is 0.
Then f(-1) = 0.

So (x + 1) is the factor of f(x).

Use the synthetic division.
(divisor's zero: -1)

The remainder is 0.
Then f(-1) = 0.

So (x + 1) is another factor of f(x).

1, -3 means (x - 3). (brown)

(x - 3) is the factor that can't be factored further.

The previously found zeros are 1, -1, and -1.

So f(x) = (x - 1)(x + 1)2(x - 3).

Example 3

Factor the given polynomial. x^4 + x^3 - 5x^2 + x - 6

Use the synthetic division.
(divisor's zero: 2)

The remainder is 0.
Then f(2) = 0.

Synthetic substitution

So (x - 2) is the factor of f(x).

Use the synthetic division.
(divisor's zero: -3)

The remainder is 0.
Then f(-3) = 0.

So (x + 3) is the factor of f(x).

1, 0, 1 means (x2 + 1). (brown)

(x2 + 1) is the factor that can't be factored further.

The previously found zeros are 2 and -3.

So f(x) = (x - 2)(x + 3)(x2 + 1).