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# Factoring polynomials by taking a common factor

Learn how to factor a common factor out of a polynomial expression. For example, factor 6x²+10x as 2x(3x+5).

#### What you should be familiar with before this lesson

The GCF (greatest common factor) of two or more monomials is the product of all their common prime factors. For example, the GCF of 6, x and 4, x, squared is 2, x.
If this is new to you, you'll want to check out our greatest common factors of monomials article.

#### What you will learn in this lesson

In this lesson, you will learn how to factor out common factors from polynomials.

## The distributive property: $a(b+c)=ab+ac$a, left parenthesis, b, plus, c, right parenthesis, equals, a, b, plus, a, c

To understand how to factor out common factors, we must understand the distributive property.
For example, we can use the distributive property to find the product of 3, x, squared and 4, x, plus, 3 as shown below:
start color #0c7f99, 3, end color #0c7f99, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, 4, x, plus, 3, right parenthesis, equals, start color #0c7f99, 3, end color #0c7f99, with, \overgroup, on top, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, 4, x, right parenthesis, plus, start color #0c7f99, 3, end color #0c7f99, with, \overgroup, on top, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, 3, right parenthesis
Notice how each term in the binomial was multiplied by a common factor of start color #0c7f99, 3, x, squared, end color #0c7f99.
However, because the distributive property is an equality, the reverse of this process is also true!
start color #0c7f99, 3, end color #0c7f99, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, 4, x, right parenthesis, plus, start color #0c7f99, 3, end color #0c7f99, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, 3, right parenthesis, equals, start color #0c7f99, 3, end color #0c7f99, with, \overgroup, on top, with, \overgroup, on top, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, 4, x, plus, 3, right parenthesis
If we start with 3, x, squared, left parenthesis, 4, x, right parenthesis, plus, 3, x, squared, left parenthesis, 3, right parenthesis, we can use the distributive property to factor out start color #0c7f99, 3, x, squared, end color #0c7f99 and obtain 3, x, squared, left parenthesis, 4, x, plus, 3, right parenthesis.
The resulting expression is in factored form because it is written as a product of two polynomials, whereas the original expression is a two-termed sum.

Problem 1
Write 2, x, left parenthesis, 3, x, right parenthesis, plus, 2, x, left parenthesis, 5, right parenthesis in factored form.

## Factoring out the greatest common factor (GCF)

To factor the GCF out of a polynomial, we do the following:
1. Find the GCF of all the terms in the polynomial.
2. Express each term as a product of the GCF and another factor.
3. Use the distributive property to factor out the GCF.
Let's factor the GCF out of 2, x, cubed, minus, 6, x, squared.
Step 1: Find the GCF
• 2, x, cubed, equals, start color #ca337c, 2, end color #ca337c, dot, start color #e07d10, x, end color #e07d10, dot, start color #e07d10, x, end color #e07d10, dot, x
• 6, x, squared, equals, start color #ca337c, 2, end color #ca337c, dot, 3, dot, start color #e07d10, x, end color #e07d10, dot, start color #e07d10, x, end color #e07d10
So the GCF of 2, x, cubed, minus, 6, x, squared is start color #ca337c, 2, end color #ca337c, dot, start color #e07d10, x, end color #e07d10, dot, start color #e07d10, x, end color #e07d10, equals, start color #0c7f99, 2, x, squared, end color #0c7f99.
Step 2: Express each term as a product of start color #0c7f99, 2, x, squared, end color #0c7f99 and another factor.
• 2, x, cubed, equals, left parenthesis, start color #0c7f99, 2, x, squared, end color #0c7f99, right parenthesis, left parenthesis, x, right parenthesis
• 6, x, squared, equals, left parenthesis, start color #0c7f99, 2, x, squared, end color #0c7f99, right parenthesis, left parenthesis, 3, right parenthesis
So the polynomial can be written as 2, x, cubed, minus, 6, x, squared, equals, left parenthesis, start color #0c7f99, 2, x, squared, end color #0c7f99, right parenthesis, left parenthesis, x, right parenthesis, minus, left parenthesis, start color #0c7f99, 2, x, squared, end color #0c7f99, right parenthesis, left parenthesis, 3, right parenthesis.
Step 3: Factor out the GCF
Now we can apply the distributive property to factor out start color #01a995, 2, x, squared, end color #01a995.
start color #0c7f99, 2, end color #0c7f99, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, x, right parenthesis, minus, start color #0c7f99, 2, end color #0c7f99, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, 3, right parenthesis, equals, start color #0c7f99, 2, end color #0c7f99, with, \overgroup, on top, with, \overgroup, on top, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, x, minus, 3, right parenthesis
Verifying our result
We can check our factorization by multiplying 2, x, squared back into the polynomial.
start color #0c7f99, 2, end color #0c7f99, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, x, minus, 3, right parenthesis, equals, start color #0c7f99, 2, end color #0c7f99, with, \overgroup, on top, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, x, right parenthesis, minus, start color #0c7f99, 2, end color #0c7f99, with, \overgroup, on top, start color #0c7f99, x, squared, end color #0c7f99, left parenthesis, 3, right parenthesis
Since this is the same as the original polynomial, our factorization is correct!

Problem 2
Factor out the greatest common factor in 12, x, squared, plus, 18, x.

Problem 3
Factor out the greatest common factor in the following polynomial.
10, x, squared, plus, 25, x, plus, 15, equals

Problem 4
Factor out the greatest common factor in the following polynomial.
x, start superscript, 4, end superscript, minus, 8, x, cubed, plus, x, squared, equals

### Can we be more efficient?

If you feel comfortable with the process of factoring out the GCF, you can use a faster method:
Once we know the GCF, the factored form is simply the product of that GCF and the sum of the terms in the original polynomial divided by the GCF.
See, for example, how we use this fast method to factor 5, x, squared, plus, 10, x, whose GCF is start color #0c7f99, 5, x, end color #0c7f99:
5, x, squared, plus, 10, x, equals, start color #0c7f99, 5, x, end color #0c7f99, left parenthesis, start fraction, 5, x, squared, divided by, start color #0c7f99, 5, x, end color #0c7f99, end fraction, plus, start fraction, 10, x, divided by, start color #0c7f99, 5, x, end color #0c7f99, end fraction, right parenthesis, equals, start color #0c7f99, 5, x, end color #0c7f99, left parenthesis, x, plus, 2, right parenthesis

## Factoring out binomial factors

The common factor in a polynomial does not have to be a monomial.
For example, consider the polynomial x, left parenthesis, 2, x, minus, 1, right parenthesis, minus, 4, left parenthesis, 2, x, minus, 1, right parenthesis.
Notice that the binomial start color #0c7f99, 2, x, minus, 1, end color #0c7f99 is common to both terms. We can factor this out using the distributive property:
x, left parenthesis, start color #0c7f99, 2, x, end color #0c7f99, start color #0c7f99, minus, 1, end color #0c7f99, right parenthesis, minus, 4, left parenthesis, start color #0c7f99, 2, x, end color #0c7f99, start color #0c7f99, minus, 1, end color #0c7f99, right parenthesis, equals, left parenthesis, x, minus, 4, right parenthesis, left parenthesis, start color #0c7f99, 2, x, minus, end color #0c7f99, with, \overgroup, on top, with, \overgroup, on top, start color #0c7f99, 1, end color #0c7f99, right parenthesis

Problem 5
Factor out the greatest common factor in the following polynomial.
2, x, left parenthesis, x, plus, 3, right parenthesis, plus, 5, left parenthesis, x, plus, 3, right parenthesis, equals

## Different kinds of factorizations

It may seem that we have used the term "factor" to describe several different processes:
• We factored monomials by writing them as a product of other monomials. For example, 12, x, squared, equals, left parenthesis, 4, x, right parenthesis, left parenthesis, 3, x, right parenthesis.
• We factored the GCF from polynomials using the distributive property. For example, 2, x, squared, plus, 12, x, equals, 2, x, left parenthesis, x, plus, 6, right parenthesis.
• We factored out common binomial factors which resulted in an expression equal to the product of two binomials. For example:
x, left parenthesis, x, plus, 1, right parenthesis, plus, 2, left parenthesis, x, plus, 1, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, plus, 2, right parenthesis
While we may have used different techniques, in each case we are writing the polynomial as a product of two or more factors. So in all three examples, we indeed factored the polynomial.

## Challenge problems

Problem 6
Factor out the greatest common factor in the following polynomial.
12, x, squared, y, start superscript, 5, end superscript, minus, 30, x, start superscript, 4, end superscript, y, squared, equals

Problem 7
A large rectangle with an area of 14, x, start superscript, 4, end superscript, plus, 6, x, squared square meters is divided into two smaller rectangles with areas 14, x, start superscript, 4, end superscript and 6, x, squared square meters.
The width of the rectangle (in meters) is equal to the greatest common factor of 14, x, start superscript, 4, end superscript and 6, x, squared.
What is the length and width of the large rectangle?
start text, W, i, d, t, h, end text, equals
meters
start text, L, e, n, g, t, h, end text, equals
meters