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 6x6x and 4x24x^2 is 2x2x.
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+aca(b+c)=ab+ac

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 3x23x^2 and 4x+34x+3 as shown below:
Notice how each term in the binomial was multiplied by a common factor of 3x2\tealD{3x^2}.
However, because the distributive property is an equality, the reverse of this process is also true!
If we start with 3x2(4x)+3x2(3)3x^2(4x)+3x^2(3), we can use the distributive property to factor out 3x2\tealD{3x^2} and obtain 3x2(4x+3)3x^2(4x+3).
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.

Check your understanding

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 2x36x22x^3-6x^2.
Step 1: Find the GCF
  • 2x3=2xxx2x^3=\maroonD2\cdot \goldD{x}\cdot \goldD{x}\cdot x
  • 6x2=23xx6x^2=\maroonD2\cdot 3\cdot \goldD{x}\cdot \goldD{x}
So the GCF of 2x36x22x^3-6x^2 is 2xx=2x2\maroonD2 \cdot \goldD x \cdot \goldD x=\tealD{2x^2}.
Step 2: Express each term as a product of 2x2\tealD{2x^2} and another factor.
  • 2x3=(2x2)(x)2x^3=(\tealD{2x^2})({x})
  • 6x2=(2x2)(3)6x^2=(\tealD{2x^2})({3})
So the polynomial can be written as 2x36x2=(2x2)(x)(2x2)(3)2x^3-6x^2=(\tealD{2x^2})( x)-(\tealD{2x^2}) ( 3).
Step 3: Factor out the GCF
Now we can apply the distributive property to factor out 2x2\tealD{2x^2}.
Verifying our result
We can check our factorization by multiplying 2x22x^2 back into the polynomial.
Since this is the same as the original polynomial, our factorization is correct!

Check your understanding

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 5x2+10x5x^2+10x, whose GCF is 5x\tealD{5x}:
5x2+10x=5x(5x25x+10x5x)=5x(x+2)5x^2+10x=\tealD{5x}\left(\dfrac{5x^2}{\tealD{5x}}+\dfrac{10x}{\tealD{5x}}\right)=\tealD{5x}(x+2)

Factoring out binomial factors

The common factor in a polynomial does not have to be a monomial.
For example, consider the polynomial x(2x1)4(2x1)x(2x-1)-4(2x-1).
Notice that the binomial 2x1\tealD{2x-1} is common to both terms. We can factor this out using the distributive property:

Check your understanding

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, 12x2=(4x)(3x)12x^2=(4x)(3x).
  • We factored the GCF from polynomials using the distributive property. For example, 2x2+12x=2x(x+6)2x^2+12x=2x(x+6).
  • We factored out common binomial factors which resulted in an expression equal to the product of two binomials. For example x(x+1)+2(x+1)=(x+1)(x+2)x(x+1)+2(x+1)=(x+1)(x+2).
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

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