How do you find the GCF of negative monomials?

Question:

How do you find the GCF of negative monomials?

Monomials:

In mathematics, a monomial is a mathematical expression that is a product of numbers, variables, and/or positive integer powers of those variables. The greatest common factor, abbreviated GCF, of a set of monomials is similar to the GCF of a set of integers, but it involves variables as well as numbers.

The GCF of a set of monomials can be found by multiplying all of the factors that are common to each of the monomial's prime factorizations together. We normally do this by finding these common factors and multiplying them together. However, when it comes to negative monomials, we have one added step. We use the following steps to find the GCF of a set of negative monomials:

1. Find the prime factorization of each of the monomials with a -1 included in that prime factorization, if necessary.
2. Identify the factors that are common to all of the prime factorizations. If -1 is a common factor of all of the prime factorizations, turn it into a positive 1 before moving on to step 3.
3. Multiply the common factors found in step 2 together. This is the GCF of the negative monomials.

For example, suppose we wanted to find the GCF of -2x2y7z and -4xy3. First, we find the prime factorization of each of the monomials with a -1 included.

• Prime factorization of -2x2y7z = -1 ⋅ 2 ⋅ xxyyyyyyyz
• Prime factorization of -4xy3 = -1 ⋅ 2 ⋅ 2 ⋅ xyyy

Now, we identify the common factors in each of these prime factorizations.

• Common factors: -1, 2, x, y, y, and y

Since -1 is in the list of common factors, we turn it positive before moving on to step 3, so we have that the common factors are 1, 2, x, y, y, and y. The third step is to multiply these together.

• 1 ⋅ 2 ⋅ xyyy = 2 ⋅ xy3 = 2xy3

We get that the GCF of -2x2y7z and -4xy3 is 2xy3, and this illustrates how to find the GCF of negative monomials.