Adding and Subtracting Polynomials

Learning Objective

• Add and subtract polynomials.

Adding and subtracting polynomialsA monomial or sum of monomials, like `4x^2 + 3x - 10`. may sound complicated, but it’s really not much different from adding and subtracting numbers. Any terms that have the same variables with the same exponents can be combined.

Adding polynomials involves combining like terms. Like termsTwo or more monomials that contain the same variables raised to the same powers, regardless of their coefficients. For example, `2x^2y` and `-8x^2y` are like terms because they have the same variables raised to the same exponents. are monomialsA number, a variable, or a product of a number and one or more variables with whole number exponents, such as `-5`, `x`, and `8xy^3`. that contain the same variable or variables raised to the same powers. The following are examples of like and unlike terms:

We combine like terms by adding or subtracting the coefficient of the term and keeping the variables and their exponents the same. The distributive property is why we can do this. Look at the example below to see that it's okay to add or subtract the coefficients of like terms:

We've just seen how to add two monomials that have like terms. We can also apply the properties of numbers when adding polynomials. To add polynomials, shuffle the expression and group like terms together to make it easier to combine them:

The procedure is the same when we are adding polynomials that contain negative coefficients or subtraction as shown below:

So far, we have been adding polynomials by reading from left to right along the same line. Some people like to organize their work vertically instead, because they find it easier to be sure that they are combining like terms. The process of adding the polynomials is the same, but the arrangement of the terms is different. The example below shows this “vertical” method of adding polynomials:

Sometimes in a vertical arrangement, we can line up every term beneath a like term, like we did in the example above. But sometimes it isn't so tidy. When there isn't a matching like term for every term, there will be empty spots in the vertical arrangement.

Subtracting polynomials also involves identifying and combining like terms. Remember that a subtraction sign in front of the parentheses is like a coefficient of `-1`. When subtracting, we can distribute (`-1`) to each of the terms in the second polynomial and then add the two polynomials. Let's look at an example:

When polynomials include a lot of terms, it can be easy to lose track of the signs. Be very careful to transfer them correctly, especially when subtracting a negative term.

As with integer operations, experience and practice makes it easier to add and subtract polynomials.

When adding or subtracting polynomials, look for like terms, which are terms that have the same variables raised to the same power. Use the commutative property of addition to regroup the terms in an expression into sets of like terms. Like terms are combined by adding or subtracting the coefficients as appropriate while keeping the variables and exponents the same.

Polynomials are not considered simplified until all of the like terms have been combined.