How to Subtract Monomials & Binomials

By Amy Harris
Students learn about special products of polynomials during high school algebra.
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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.