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.
More Algebra Lessons
More Algebra Worksheets
More Algebra Games
What is the Factor Theorem?
) is divided by x
, we get
f(x) = (x – a)Q(x) + remainder
From the Remainder Theorem
, we get
f(x) = (x – a)Q(x) + f(a)
) = 0 then the remainder is 0 and
f(x) = (x – a)Q(x)
We can then say that x
is a factor of f(x
The Factor Theorem
x – a 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 + x3 – x2 + 3x + 2
b) x6 + 2x(x – 1) – 4
a) Let f(x) = 3x4 + x3 – x2 + 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?
What the theorems are and how they can be used to find the linear factorization of a polynomial?
The Remainder Theorem states that if a polynomial, f(x), is divided by x - k, the remainder is equal to f(k).
The Factor Theorem states that the polynomial x - k is a factor of the polynomial f(x) if and only if f(k) = 0.
Let f(x) = 2x3
− 5x + 6
Is x - 1 a factor?
Find all the other factors.
How to use the Factor Theorem to factor polynomials?
1) Factor P(x) =
− 19x + 8
2) Factor P(x) =
+ x + 12
How to find remaining factors of a polynomial?
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.
3. To use synthetic division, along with the factor theorem to help factor a polynomial.
Fully factor x4
+ 15x + 18
Application of the Factor Theorem
How to use the factor theorem to determine if x - c is a factor of f. If it is factor the polynomial?
1. f(x) = 4x3
- 8x + 4, c = 3
2. f(x) = 3x4
- 5x + 10, c = 1
3. f(x) = 3x6
- 176, c = -2
4. f(x) = 4x6
- 16, c = 4
5. f(x) = 2x4
- 2x - 1, c = -1/2
How to explain 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)
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
− x − 3?
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.
You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.