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Factor Theorem

In these lessons, we will look at the Factor Theorem and how it relates to the Remainder Theorem. We will also show how to factor polynomials using the Factor Theorem.

Related Topics:
More Algebra Lessons, More Algebra Worksheets, More Algebra Games

What is the Factor Theorem?
When f(x) is divided by xa, we get

f(x) = (xa)Q(x) + remainder
From the Remainder Theorem, we get
f(x) = (xa)Q(x) + f(a)
If f(a) = 0 then the remainder is 0 and
f(x) = (xa)Q(x)
We can then say that xa is a factor of f(x)
The Factor Theorem states that
xa is a factor of the polynomial f(x) if f(a) = 0


Determine whether x + 1 is a factor of the following polynomials.

a) 3x4 + x3x2 + 3x + 2

b) x6 + 2x(x – 1) – 4


a) Let f(x) = 3x4 + x3x2 + 3x + 2

f(–1) = 3(–1)4 + (–1)3 – (–1)2 +3(–1) + 2

= 3(1) + (–1) – 1 – 3 + 2 = 0

Therefore, x + 1 is a factor of f(x)

b) Let g(x) = x6 + 2x(x – 1) – 4

g(–1) = (–1)6 + 2(–1)( –2) –4 = 1

Therefore, x + 1 is not a factor of g(x)

How to use the Factor Theorem and Remainder Theorem?
The Factor Theorem and The Remainder Theorem
What the theorems are and how they can be used to find the linear factorization of a polynomial?
Factor F(x) = 2x3 − 3x2 − 5x + 6
This video demonstrates how to use the Factor Theorem to factor polynomials.
1) Factor P(x) = 3x3 − x2 − 19x + 8
1) Factor P(x) = 2x3 − 9x2 + x + 12

The Factor Theorem
If f(x) is a polynomial and f(p) = 0 then (x − p) is a factor of f(x)
If f(x) is a polynomial and f(−q) = 0 then (x + q) is a factor of f(x)
Polynomials - Factor Theorem
Description and examples of the Factor Theorem
Prove that (x + 1) is a factor of P(x) = x2 + 2x + 1
Is (x + 2) a factor of x3 + 4x2 − x − 3?

The Factor Theorem
A lesson on the factor theorem and completely factoring a polynomial.
1. To learn the connection between the factor theorem and the remainder theorem
2. To learn how to use the factor theorem to determine if a binomial is a factor of a given polynomial or not.
Fully factor x4 − 3x3 − 7x2 + 15x + 18
Dividing Polynomials; Remainder and Factor Theorem.


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