Greatest common factor of monomials

Learn how to find the GCF (greatest common factor) of two monomials or more.

What you should be familiar with before this lesson

A monomial is an expression that is the product of constants and nonnegative integer powers of xx, like 3x23x^2. A polynomial is a sum of monomials.
You can write the complete factorization of a monomial by writing the prime factorization of the coefficient and expanding the variable part. Check out our Factoring monomials article if this is new to you.

What you will learn in this lesson

In this lesson, you will learn about the greatest common factor (GCF) and how to find this for monomials.

Review: Greatest common factors in integers

The greatest common factor of two numbers is the greatest integer that is a factor of both numbers. For example, the GCF of 1212 and 1818 is 66.
We can find the GCF for any two numbers by examining their prime factorizations:
  • 12=22312=\blueD{2}\cdot 2\cdot \goldD{3}
  • 18=23318=\blueD{2}\cdot \goldD3\cdot 3
Notice that 1212 and 1818 have a factor of 2\blueD{2} and a factor of 3\goldD{3} in common, and so the greatest common factor of 1212 and 1818 is 23=6\blueD{2}\cdot \goldD{3}=6.

Greatest common factors in monomials

The process is similar when you are asked to find the greatest common factor of two or more monomials.
Simply write the complete factorization of each monomial and find the common factors. The product of all the common factors will be the GCF.
For example, let's find the greatest common factor of 10x310x^3 and 4x4x:
  • 10x3=25xxx10x^3=\blueD{2}\cdot 5\cdot \goldD{x}\cdot x\cdot x
  • 4x=22x4x=\blueD{2}\cdot 2\cdot \goldD{x}
Notice that 10x310x^3 and 4x4x have one factor of 2\blueD{2} and one factor of x\goldD{x} in common. Therefore, their greatest common factor is 2x\blueD2\cdot \goldD{x} or 2x2x.

Check your understanding

1) What is the greatest common factor of 9x29x^2 and 6x6x?
Choose 1 answer:
Choose 1 answer:

Let's factor each monomial completely. Then, we can find the factors common to both monomials and multiply them to find the GCF.
  • 9x2=33xx9x^2=\blueD{3}\cdot {3}\cdot \goldD{x}\cdot x
  • 6x=23x6x=2 \cdot \blueD{3}\cdot\goldD{x}
Notice that each monomial has one factor of 3\blueD3 and one factor of x\goldD{x}. Therefore, the greatest common factor of the monomials is 3x\blueD3 \cdot \goldD{x} or 3x3x
2) What is the greatest common factor of 12x512x^5 and 8x38x^3?

Let's factor each monomial completely. Then, we can find the factors common to both monomials and multiply them to find the GCF.
  • 12x5=223xxxxx12x^5=\blueD{2}\cdot \blueD{2} \cdot {3}\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x} \cdot x \cdot x
  • 8x3=222xxx8x^3=\blueD2 \cdot \blueD{2}\cdot 2 \cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}
Notice that each monomial has two factors of 2\blueD2 and three factors of x\goldD{x}. Therefore, the greatest common factor of the monomials is 22xxx\blueD2 \cdot \blueD2 \cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x} or 4x34x^3.
3) What is the greatest common factor of 5x75x^7, 30x430x^4, and 10x310x^3?

Let's factor each monomial completely. Then, we can find the factors common to all three monomials and multiply them to find the GCF.
  • 5x7=5xxxxxxx5x^7=\blueD5\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}\cdot x\cdot x\cdot x\cdot x
  • 30x4=235xxxx30x^4=2\cdot {3}\cdot \blueD{5}\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}\cdot x
  • 10x3=25xxx10x^3= {2}\cdot \blueD5\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}
Notice that each monomial has one factor of 5\blueD{5} and three factors of x\goldD{x}. Therefore, the greatest common factor of the monomials is 5xxx\blueD5\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x} or 5x35x^3.

A note on the variable part of the GCF

In general, the variable part of the GCF for any two or more monomials will be equal to the variable part of the monomial with the lowest power of xx.
For example, consider the monomials 6x5\blueD{6}\goldD{x^5} and 4x2\blueD{4}\goldD{x^2}:
  • Since the lowest power of xx is x2\goldD{x^2}, that will be the variable part of the GCF.
  • You could then find the GCF of 6\blueD6 and 4\blueD4, which is 2\blueD2, and multiply this by x2\goldD{x^2} to obtain 2x2\blueD2\goldD{x^2}, the GCF of the monomials!
Sure! The complete factorization of each monomial is given below:
  • 6x5=23xxxxx6x^5=\blueD2\cdot 3\cdot \goldD{x}\cdot \goldD{x}\cdot x\cdot x\cdot x
  • 4x2=22xx4x^2=\blueD2\cdot 2\cdot \goldD{x}\cdot\goldD{x}
Notice that the GCF is indeed 2x22x^2.
This is especially helpful to understand when finding the GCF of monomials with very large powers of xx. For example, it would be very tedious to completely factor monomials like 32x10032x^{100} and 16x8816x^{88}!

Challenge Problems

4*)What is the greatest common factor of 20x7620x^{76} and 8x92 8x^{92}?

Notice that 20x7620x^{76} is the monomial with the lowest power of xx. So the variable part of the GCF will be x76x^{76}.
The GCF of 2020 and 88 is 44.
Therefore, the GCF of 20x7620x^{76} and 8x928x^{92} is 4x764x^{76}.
5*) What is the greatest common factor of 40x5y240x^5y^2 and 32x2y3 32x^2y^3?

To find the greatest common factor of 40x5y240x^5y^2 and 32x2y332x^2y^3, let's factor each monomial completely, and see what is common.
  • 40x5y2=2225xxxxxyy40x^5y^2=\blueD2\cdot \blueD2\cdot \blueD{2}\cdot 5\cdot \goldD{x}\cdot \goldD{x}\cdot {x}\cdot x\cdot x \cdot\greenD{y}\cdot \greenD{y}
  • 32x2y3=22222xxyyy32x^2y^3=\blueD{2}\cdot \blueD{2}\cdot\blueD{2}\cdot{2}\cdot {2}\cdot \goldD{x}\cdot \goldD{x}\cdot\greenD{y}\cdot \greenD{y}\cdot y
The monomials each have three factors of 2\blueD{2}, two factors of x\goldD{x}, and two factors of y\greenD{y} in common. Therefore, the greatest common factor of the polynomial is 222xxyy\blueD2\cdot\blueD2\cdot \blueD 2 \cdot \goldD{x}\cdot \goldD{x}\cdot \greenD{y}\cdot \greenD{y} or 8x2y28x^2y^2.
Alternatively, notice that the lowest power of xx is x2x^2 and the lowest power of yy is y2y^2. So the variable part of the GCF will be x2y2x^2y^2. The GCF of 4040 and 3232 is 88, and so the GCF of the monomials is 8x2y28x^2y^2.

What's next?

To see how we can use these skills to factor polynomials, check out our next article on factoring out the greatest common factor!
Loading