Factoring
Polynomial
After going through this module, you are
expected to factor polynomials (using the
Re m a i n d e r T h e o r e m a n d t h e Fa c t o r
Theorem)
To determine the factor of a polynomial
you can use Factor Theorem, Synthetic
Division, or Remainder Theorem. Let us
consider the given polynomial on your
previous activity.
What is the Factor Theorem?
When f(x) is divided by (x – a), we get
f(x) = (x – a)Q(x) + remainder
From the Remainder Theorem, we get
f(x) = (x – a)Q(x) + f(a)
If f(a) = 0 then the remainder is 0 and
f(x) = (x – a)Q(x)
We can then say that (x – a) is a factor of f(x)
The Factor Theorem states that
(x – a) is a factor of the polynomial f(x) if and only if f(a) = 0
Take note that the following statements are equivalent for any polynomial f(x).
• (x – a) is a factor of f(x).
• The remainder is zero when f(x) is divided by (x – a).
• f(a) = 0.
• The solution to f(x) = 0 is a.
• The zero of the function f(x) is a.
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.
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)
Illustrative Example 1
What are the factors of x2 + 6x + 8?
Using the remainder theorem and x + 2 (one of the choices in column B of the previous
activity) as your divisor, x = -2
P(x) = x2 + 6x + 8
P(-2) = (-2)2 + 6(-2) + 8
P(-2) = 4 – 12 + 8
P(-2) = 0
So, x + 2 is indeed a factor of x2 + 6x + 8.
The value of c should be taken from the constant 8. The factors of 8 are 1, 2, 4, 8 either both
positive or negative. Since the polynomial is of degree to then there are two factors.
Now, use long division or synthetic division to get the other factor. Since -2 was used as the
value of , then the other factor that we can multiplied by –2 that gives the product of 8 is –4.
Using Synthetic Division:
- 41 6 8
-4 -8
1 2 0
This means the other factor is x + 4. So, the factors of
x2 + 6x + 8 are x+2 and x + 4.
Illustrative Example 2
�3 – 7x + 6
Try x – 1 as one of the factors.
Using synthetic division:
1 1 0 -7 6
1 1 -6
1 1 -6 0
Since the remainder is zero, then x - 1 is one of the factors of �3 – 7x + 6. What
are the other factors of the polynomial?
The resulting expression or quotient in the synthetic division is �2 + x - 6 , you can
now find the factors of this expression by using your factoring skills or by trial and
error. Think of two numbers that have a product of -6 and the sum of 1.
Those numbers are 3 and -2.
So, �2 - + x – 6 = (x + 3)(x - 2).
Therefore, the factors of the polynomial
�3 – 7x + 6 are (x - 1)(x + 3)(x - 2).
A third degree polynomial has at most 3 factors.
Illustrative Example 3
What are the factors of �3 − 2�2 – 5x + 6?
Choosing x – 3 as one of the factors, we do the synthetic division:
3 1 -2 -5 6
3 3 -6
1 1 -2 0
since the remainder is zero, then x - 3 is one of the factors of �3 − 2�2 – 5x + 6. What are the
other factors of the polynomial?
The resulting expression or quotient in the synthetic division is �2 + x -2, you can now find the
factors of this expression by using your factoring skills or by trial and error. Think of two numbers
that have a product of -2 and the sum of 1.
Therefore, the factors of the polynomial ��
− ��� – 5x + 6 are (x - 3)(x + 2)(x - 1).
It is now your turn to
answer your Learning
Tasks #1, 2 and 3