Advanced Understanding of Like Terms


Before you can do any operations on polynomials, you have to be able to identify what like terms are. But before we can do that, you have to know what terms are first. Consider the following polynomial:

4ab + a^2b^3 - 3b^2 + 1 

Terms are individual parts of a polynomial added together. So in the polynomial above, the terms are as follows:

4ab 
a^2b^3 
-3b^2 
1 

Each term has two parts, the coefficient portion and the variable portion. The coefficient is the numerical constant part of a term, and the variable portion is all of the variables multiplied together. So for example, here are the four terms above broken up into the coefficient portion and the variable portion, separated by parentheses with the coefficient portion listed first:

4ab = (4)(ab) 
a^2b^3 = (1)(a^2b^3) 
-3b^2 = (-3)(b^2) 
1 = (1) 

Notice that the second term listed has a coefficient of 1, but it’s not written because it’s understood. Similarly, when we write a simple equation like x + 3 = 5, the coefficient of the x term is 1, but we don’t write it because we know that x means 1 * x. It’s also worth noting that in the third term we keep the negative portion with the coefficient instead of the variables. So if we have a term of -5x, that would mean the coefficient portion of the term is -5 and the variable portion is x.

Now that you understand what terms are, what are like terms? Like terms are terms that have the same variable portion. Here is an example polynomial to show you exactly what we mean:

2ab + 3a^2b - 4ab^2 - 5a^2b^2 + 6a^2b + 7a^2b^2 + 8a^2b + 9ab 

Let’s start with the term 2ab. The coefficient is 2 and the variable portion is ab. As we look across this polynomial, we see that the term 9ab also has a variable portion of ab. Therefore, 2ab and 9ab are like terms. Note that 2ab and 3a²b are not like terms because their variable portions are ab and a²b, which are not the same. Both the variables themselves and the exponents the variables are raised to have to be the same before two terms are considered like terms.

Here’s one more example to tackle a common mistake students make:

2x^2y + 3xy^2 + 5y^2x + 6yx^2 

Here you might need to ask if there are any like terms in this polynomial? The answer is actually yes, but it’s hard to see it at first.

Consider 2x²y and 6yx². The variable portions of these two terms are x²y and yx², respectively. It’s important to remember that multiplication is commutative, and that as a result we have x²y = yx². To avoid this kind of confusion, you should write the variables in your terms in alphabetical order. This makes it easier to see like terms by just glancing at a polynomial. For example, the term 2b³x²wac² should be written as 2ab³c²wx², keeping the variables in alphabetical order. Just don’t forget that the exponents stay with their variables!