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.
Example: Consider 10x²y3 and 15xy²
The greatest common factor is 5xy² .
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 y² .
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.
Example 1: 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)
Example 2: Factor: 15x2y3 + 10xy²
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 y².
The GCF is 5xy². Divide each term by the GCF.
Answer: 5xy²(3xy + 2)