Step-by-step explanation:
A monomial is an expression that is the product of constants and nonnegative integer powers of xx, like 3x^23x
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.
In this lesson, you will learn about the greatest common factor (GCF) and how to find this for monomials.
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=\blueD{2}\cdot 2\cdot \goldD{3}12=2⋅2⋅3
18=\blueD{2}\cdot \goldD3\cdot 318=2⋅3⋅3
Here is an example:
What is the greatest common factor of 9x^29x
2
and 6x6x?
The anwser is 3x
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 10x^310x
3
and 4x4x:
10x^3=\blueD{2}\cdot 5\cdot \goldD{x}\cdot x\cdot x10x
3
=2⋅5⋅x⋅x⋅x
4x=\blueD{2}\cdot 2\cdot \goldD{x}4x=2⋅2⋅x
Notice that 10x^310x
3
and 4x4x have one factor of \blueD{2}2 and one factor of \goldD{x}x in common. Therefore, their greatest common factor is \blueD2\cdot \goldD{x}2⋅x or 2x2x.
I think you dont understand that so i will help you more
Q: What is the greatest common factor of 5x^75x
7
, 30x^430x
4
, and 10x^310x
3
?
A:Let's factor each monomial completely. Then, we can find the factors common to all three monomials and multiply them to find the GCF.
5x^7=\blueD5\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}\cdot x\cdot x\cdot x\cdot x5x
7
=5⋅x⋅x⋅x⋅x⋅x⋅x⋅x
30x^4=2\cdot {3}\cdot \blueD{5}\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}\cdot x30x
4
=2⋅3⋅5⋅x⋅x⋅x⋅x
10x^3= {2}\cdot \blueD5\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}10x
3
=2⋅5⋅x⋅x⋅x
Notice that each monomial has one factor of \blueD{5}5 and three factors of \goldD{x}x. Therefore, the greatest common factor of the monomials is \blueD5\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}5⋅x⋅x⋅x or 5x^35x
3
.
Hope i helped you and i know its really really large
