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.
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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?
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