How to Subtract Monomials & Binomials

By Amy Harris
Operations on monomials and binomials are taught in high school algebra.

Monomials and binomials are both types of algebraic expressions. Monomials possess one single term, as is the case in 6x^2, while binomials possess two terms separated by a plus or minus sign, as in 6x^2 – 1. Both monomials and binomials can consist of variables, with their exponents and coefficients, or constants. A coefficient is a number appearing on the left side of a variable that is multiplied by the variable; for example, in the monomial 8g, “eight” is a coefficient. A constant is a number without an attached variable; for example, in the binomial -7k + 2, “two” is a constant.

Subtracting Two Monomials

Ensure that the two monomials are like terms. Like terms are terms possessing the same variables and exponents. For instance, 7x^2 and -4x^2 are like terms, since they both share the same variable and exponent, x^2. But 7x^2 and -4x aren’t like terms because their exponents differ, and 7x^2 and -4y^2 aren’t like terms because their variables differ. Only like terms can be subtracted.

Subtract the coefficients. Consider the problem -5j^3 – 4j^3. Subtracting the coefficients, -5 – 4, produces -9.

Write the resulting coefficient to the left of the variable and exponent, which remain unchanged. The previous example yields -9j^3.

Subtracting One Monomial and One Binomial

Rearrange the terms so that like terms appear next to each other. For instance, suppose you are asked to subtract the monomial 4x^2 from the binomial 7x^2 + 2x. In this case, the terms are initially written 7x^2 + 2x – 4x^2. Here, 7x^2 and -4x^2 are like terms, so reverse the last two terms, putting the 7x^2 and -4x^2 next to each other. Doing so yields 7x^2 – 4x^2 + 2x.

Perform subtraction on the coefficients of the like terms, as described in the previous section. Subtract 7x^2 – 4x^2 to get 3x^2.

Write this result along with the remaining term from Step 1, which in this case is 2x. The solution to the example is 3x^2 + 2x.

Subtracting Two Binomials

Use the distributive property to change subtraction to addition when there are parentheses involved. For instance, in 8m^5 – 3m^2 – (6m^5 – 9m^2), distribute the minus sign appearing to the left of the parentheses to both terms inside the parentheses, 6m^5 and -9m^2 in this case. The example becomes 8m^5 – 3m^2 – 6m^5 – -9m^2.

Change any minus signs appearing directly next to negative signs into a single plus sign. In 8m^5 – 3m^2 – 6m^5 – -9m^2, a minus sign appears next to a negative in between the last two terms. These signs become a plus sign, and the expression becomes 8m^5 – 3m^2 – 6m^5 + 9m^2.

Reorder the terms so that like terms are grouped next to each other. The example becomes 8m^5 – 6m^5 – 3m^2 + 9m^2.

Combine like terms by adding or subtracting as indicated in the problem. In the example, subtract 8m^5 – 6m^5 to get 2m^5, and add -3m^2 + 9m^2 to get 6m^2. Put these two results together for a final solution of 2m^5 + 6m^2.

About the Author

Based in western New York, Amy Harris began writing for Demand Media and Great Lakes Brewing News in 2010. Harris holds a Bachelor of Science in Mathematics from Penn State University; she taught high school math for several years and has also worked in the field of instructional design.