When
two integers are multiplied together, the answer is called a product.
The integers that were multiplied together are called the factors
of the product.
3
• 6 = 18
(3 and 6 are factors of 18)



The greatest
common factor of two (or more) integers is the largest integer
that is a factor of both (or all) numbers.
Consider the
numbers 18, 24, and 36.
The greatest common factor is 6.
(6 is the largest integer
that will divide evenly into all three numbers)


The greatest
common factor, (GCF), of two (or more) monomials is the product of the
greatest common factor of the numerical coefficients (the numbers out in
front) and the highest power of every variable that is a factor of each
monomial.

Consider:
10x^{2}y^{3} and 15xy^{2 }
The greatest common factor is 5xy^{2} .
The largest factor of 10
and 15 is 5.
The highest power of x that is contained in both
terms is x.
The highest power of y that is contained in both
terms is y^{2} . 

When factoring
polynomials, first look for the largest monomial which is a factor of
each term of the polynomial. Factor
out (divide each term by) this largest monomial.
Factor: 4x + 8y
The largest integer that will divide evenly
into 4 and 8 is 4. Since the terms do not contain a variable
(x or y) in common, we cannot factor any variables.
The greatest common factor is 4. Divide each term by
4.
Answer: 4(x + 2y) 


Factor: 15x^{2}y^{3} + 10xy^{2
}The largest integer that will divide evenly into 15 and 10
is 5. The largest power of x present in both terms is
x.
The largest power of y present in both term is y^{2}.
The GCF is 5xy^{2}. Divide each term by
the GCF.
Answer: 5xy^{2}(3xy
+ 2) 


