Factoring Polynomials Out the Greatest Common Factor
This chapter describes how to factor algebraic expressions with the greatest common factor (gcf). There are several different ways to examine in detail one at a time, depending on the expression we want to factor. The exercises available in this chapter will allow you to apply each of the methods studied.
By this definition, the algebraic expression must consist of at least two terms in order to find the common factor between these terms.
We take a simple examples of algebraic expressions with the number of their terms:
In , we have two terms: 18 and .
In , we have two terms: and .
In , we have three terms: , and .
Info!
Factoring is useful in several cases, including solving equations and studying functions.
Factoring with a common factor: esay case
In this lesson we will need the greatest common factor, so I invite you to go to Lesson hwo to find gcf before you read on here.
taking simple examples to fully understand:
In , the common factor is 5.
In , the common factor is .
In , the common factor is .
In , the common factor is .
Now we know what the common factor is and how to define it in an algebraic expression. Now let's work out simple algebraic expressions where the common factor is a simple number or variable.
Example1:
Factor the following expressions:
Solution:
- In , the common factor is 7, so:
- In , the common factor is 7, so:
- In , the common factor is 2, so:
- In , the common factor is , so:
- In , the common factor is , so:
- In , the common factor is , so:
As you can see, the factor is put in front, then in the parenthesis we rewrite the starting expression without the common factor.
To check, we can take the expression on the right and expand it using the distributive property, we should find the expression on the left:
Factoring with a not obvious greatest common factor
there are some cases where the common factor is not obvious, it has to be brought out.
For example, the expression we don't see the common factor, but we can make it appear by decomposing the terms.
Therefore, the purpose of this paragraph is to rewrite the expression in a different way in order to show the common factor.
Let's back to the privious expression, can be rewritten in two different ways:
In the fisrt line, the common factor is 2, in the second line, the common factor is 4. In this case, always, we choose the greatest common factor, that's mean we factor by 4.
Note that there are common factors may be a bit more complicated to find it.
Example2:
Factor the following expressions:
Solution:
- Its obvious that , therefore:
- Its obvious that gcf(4;6)=2, and is another common factor, therefore, the greatest common factor is :
- In , its clear that , and is another common factor, therefore, the greatest common factor is :
- In , simply, the common factor is , therefore:
- What do you think about this expression? Can you find the common factor? :
Factoring polynomials by grouping
In this section, we will try to factor polynomials that containes parenthesis or brackets and the common factor will be the expression between the parenthesis. We apply the reciprocal property in the same way as above, but this time we call it factoring by grouping.
For example, look closely at this expression: . You've no doubt noticed that the common factor is . So,
Example3:
Factor the following expressions:
Solution:
Warning!
You may be need to be familiar with adding and subtracting the polynomials. I suggest you read this lesson.
- The common factor is :
- The common factor is :
- The common factor is :
- The common factor is :
- The common factor is , right? Oh no!! look good at the expression, we have , so, the common factor is :