Mathwords: Factor Theorem

Q: What is the difference between the Factor Theorem and the Remainder Theorem?

The Remainder Theorem states that when you divide $p(x)$ by $(x - c)$, the remainder equals $p(c)$. The Factor Theorem is a special case: when that remainder is zero ($p(c) = 0$), then $(x - c)$ divides $p(x)$ exactly, making it a factor. So the Factor Theorem adds the word 'factor' to the Remainder Theorem's conclusion about remainders.

Q: How do you use the Factor Theorem to factor a polynomial completely?

First, test candidate values of $c$ (often using the Rational Root Theorem to find likely candidates). When you find a $c$ where $p(c) = 0$, you know $(x - c)$ is a factor. Divide $p(x)$ by $(x - c)$ using synthetic or long division to get a quotient polynomial of lower degree. Repeat the process on the quotient until you have factored completely.

Key Formula

\text{If } p(x) \text{ is a polynomial, then } (x - c) \text{ is a factor of } p(x) \iff p(c) = 0

Where:

$p(x)$ = A polynomial in the variable x
$c$ = A constant; a candidate zero of the polynomial
$(x - c)$ = A linear binomial that may be a factor of p(x)

Worked Example

Problem: Show that

(x - 2)

is a factor of

p(x) = x^3 - 6x^2 + 11x - 6

.

Step 1: The Factor Theorem says

(x - 2)

is a factor if and only if

p(2) = 0

. Substitute

x = 2

into the polynomial.

p(2) = (2)^3 - 6(2)^2 + 11(2) - 6

Step 2: Evaluate each term.

p(2) = 8 - 24 + 22 - 6

Step 3: Combine the values.

p(2) = 0

Step 4: Since

p(2) = 0

, by the Factor Theorem,

(x - 2)

is a factor of

p(x)

. You can verify by dividing:

x^3 - 6x^2 + 11x - 6 = (x - 2)(x^2 - 4x + 3) = (x - 2)(x - 1)(x - 3)

Answer: Because

p(2) = 0

, the Factor Theorem confirms that

(x - 2)

is a factor of

x^3 - 6x^2 + 11x - 6

.

Another Example

This example shows a negative candidate value ( $c = -3$ ) and demonstrates what happens when the test fails — the binomial is not a factor.

Problem: Determine whether

(x + 3)

is a factor of

p(x) = 2x^3 + 3x^2 - 11x + 6

.

Step 1: Rewrite

(x + 3)

in the form

(x - c)

. Here

c = -3

.

x + 3 = x - (-3)

Step 2: Evaluate

p(-3)

by substituting

x = -3

.

p(-3) = 2(-3)^3 + 3(-3)^2 - 11(-3) + 6

Step 3: Compute each term:

2(-27) = -54

,

3(9) = 27

,

-11(-3) = 33

.

p(-3) = -54 + 27 + 33 + 6 = 12

Step 4: Since

p(-3) = 12 \neq 0

, the Factor Theorem tells us

(x + 3)

is NOT a factor of

p(x)

.

Answer:

(x + 3)

is not a factor of

2x^3 + 3x^2 - 11x + 6

because

p(-3) = 12 \neq 0

.

Frequently Asked Questions

What is the difference between the Factor Theorem and the Remainder Theorem?

The Remainder Theorem states that when you divide

p(x)

by

(x - c)

, the remainder equals

p(c)

. The Factor Theorem is a special case: when that remainder is zero (

p(c) = 0

), then

(x - c)

divides

p(x)

exactly, making it a factor. So the Factor Theorem adds the word 'factor' to the Remainder Theorem's conclusion about remainders.

How do you use the Factor Theorem to factor a polynomial completely?

First, test candidate values of

c

(often using the Rational Root Theorem to find likely candidates). When you find a

c

where

p(c) = 0

, you know

(x - c)

is a factor. Divide

p(x)

by

(x - c)

using synthetic or long division to get a quotient polynomial of lower degree. Repeat the process on the quotient until you have factored completely.

Does the Factor Theorem work for all polynomials?

Yes, the Factor Theorem applies to any polynomial with real or complex coefficients. It works for polynomials of any degree. However, it only addresses linear factors of the form

(x - c)

. Irreducible quadratic factors (over the reals) require other techniques to identify.

Factor Theorem vs. Remainder Theorem

	Factor Theorem	Remainder Theorem
Statement	$(x - c)$ is a factor of $p(x)$ if and only if $p(c) = 0$	When $p(x)$ is divided by $(x - c)$ , the remainder is $p(c)$
What it tells you	Whether a specific binomial is a factor (yes/no)	The exact numerical remainder of polynomial division
Relationship	Special case of the Remainder Theorem (remainder = 0)	The more general result; applies even when the remainder is nonzero
Typical use	Testing whether a value is a root; factoring polynomials	Finding remainders without performing full division

Why It Matters

The Factor Theorem is one of the most frequently used tools in algebra and precalculus for factoring higher-degree polynomials. It connects two fundamental ideas — roots and factors — letting you move freely between solving equations and rewriting expressions in factored form. You will rely on it when graphing polynomials, solving polynomial equations, and proving divisibility results in later courses.

Common Mistakes

Mistake: Confusing the sign of

c

when the factor is

(x + c)

. For example, testing

p(3)

when checking whether

(x + 3)

is a factor.

Correction: Always rewrite the factor as

(x - c)

first. Since

(x + 3) = (x - (-3))

, you must evaluate

p(-3)

, not

p(3)

.

Mistake: Concluding that

c

is a factor instead of

(x - c)

. Students sometimes say '2 is a factor' when they mean

(x - 2)

is a factor.

Correction: The number

c

is a zero (root) of the polynomial. The factor is the binomial

(x - c)

. Keep the language precise: zeros are numbers, factors are expressions.

Related Terms

Factor of a Polynomial — The expression $(x - c)$ identified by the theorem
Zero of a Function — The value $c$ where $p(c) = 0$
Polynomial — The type of function the theorem applies to
Theorem — General term for a proven mathematical statement
Polynomial Facts — Collection of key polynomial properties and results
Remainder Theorem — The general theorem of which Factor Theorem is a special case
Synthetic Division — Efficient division method used alongside the theorem
Rational Root Theorem — Helps identify candidate values of $c$ to test