Greatest Common Factor of Two Monomials

(Slice: Polynomials and Quadratic Equations)

Recall that a monomial consists of a product of numbers and variables to positive powers. When you string several monomials together with + and – signs between them, you have a polynomial and the monomials that make it up are called terms.

This objective concerns looking for the largest factor we can divide out of two or more monomials. The thought process is very similar to reducing fractions. To reduce 9/12 we would divide the top and bottom by the biggest number that goes evenly into both, which is 3. This would give us a reduced fraction of ¾.

To find the greatest common factor of two monomials first consider the coefficients (number in front of variables) using the same thought process you would use to reduce a fraction…what is the biggest number that divides into both?

Example:

Find the greatest common factor of the two expressions:

and

The largest number that divides evenly into 4 and 20 is 4. It will be part of the greatest common factor.

Next we look at the variables one by one.

We have a t to the 8th power and a t to the 5th power. The largest one that divides into both of these is the smallest exponent, which is 5. Thus, t to the fifth power is part of the greatest common factor. 

Think for a minute about why the smallest exponent is the greatest common factor so the use of greatest doesn't get you confused.  If you are looking for something to divide into two or more numbers, it cannot be any bigger than the smallest of the numbers.  If it were, how would you divide a bigger number into a smaller one?  That's how we get fractions...something we don't want here.

We also have u in both monomials. Since 5 is the smaller of the two exponents on the u variables, u to the fifth will be in our greatest common factor.

Only one term has a z so it can’t be in a common (meaning in common to all of the monomials you are working with) factor.

Thus the greatest common factor for and is .