12/5/2016
(67) Greatest common factor of monomials | Common monomial factors | Polynomial factorization | Algebra I | Khan Academy
Greatest common factor of monomials
Learn how to find the GCF (greatest common factor) of two monomials or more.
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What you should be familiar with before this lesson
A monomial is an expression that is the product of constants and
2
nonnegative integer powers of x, like 3x . 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 12 and 18 is 6.
We can find the GCF for any two numbers by examining their prime
factorizations:
12 = 2 ⋅ 2 ⋅ 3
18 = 2 ⋅ 3 ⋅ 3
https://www.khanacademy.org/math/algebra/polynomial-factorization/common-monomial-factors/a/greatest-common-factor-of-monomials
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12/5/2016
(67) Greatest common factor of monomials | Common monomial factors | Polynomial factorization | Algebra I | Khan Academy
Notice that 12 and 18 have a factor of 2 and a factor of 3in common, and so
the greatest common factor of 12 and18 is 2 ⋅ 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.
3
For example, let's find the greatest common factor of 10x and 4x:
3
10x = 2 ⋅ 5 ⋅ x ⋅ x ⋅ x
4x = 2 ⋅ 2 ⋅ x
3
Notice that 10x and 4x have one factor of 2 and one factor of x in common.
Therefore, their greatest common factor is 2 ⋅ x or 2x.
Check your understanding
2
1) What is the greatest common factor of 9x and 6x?
2
3x
3x
18x
2
18x
2
54x
Check
[I need help!]
2) What is the greatest common factor
5
3
of 12x and8x ?
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12/5/2016
(67) Greatest common factor of monomials | Common monomial factors | Polynomial factorization | Algebra I | Khan Academy
Check
[I need help!]
7
4
3) What is the greatest common factor of 5x , 30x ,
3
and 10x ?
Check
[I need help!]
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 x.
5
2
For example, consider the monomials 6x and 4x :
2
Since the lowest power of x is x , that will be the variable part of the GCF.
You could then find the GCF of 6 and 4, which is 2, and multiply this
2
2
by x to obtain 2x , the GCF of the monomials!
The lowest power of
5
2
2
x and x is x .
5
2
2
GCF(6 x , 4 x ) = 2 x
The GCF of 6 and 4 is 2.
[I'd like to verify this using the factorization method.]
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12/5/2016
(67) Greatest common factor of monomials | Common monomial factors | Polynomial factorization | Algebra I | Khan Academy
This is especially helpful to understand when finding the GCF of monomials
with very large powers of x. For example, it would be very tedious to
100
completely factor monomials like 32x
88
and 16x !
Challenge Problems
4*)What is the greatest common factor
76
92
of 20x and8x ?
Check
[I need help!]
5*) What is the greatest common factor
5 2
of 40x
2 3
y and32x y ?
Check
[I need help!]
https://www.khanacademy.org/math/algebra/polynomial-factorization/common-monomial-factors/a/greatest-common-factor-of-monomials
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