Simplifying Expressions

There are many ways to represent a given value mathematically. For instance, the number 8 has the same value as the expression , which has the same value as the expression , etc. In fact, a given value can be represented in infinitely many ways. However, representing the value as 8 is far simpler than .

In many circumstances in mathematics it becomes important to express a value in its simplest terms. In order to do that, we must first have an idea of what is meant by “like terms.” In other words, we need to identify values that can be combined.

Suppose that we wanted to add the expressions and . We would need to identify which terms can naturally be added to one another. In this case, certainly the 6 and the 8, both being numbers, can be added together to get 14. It’s a little more difficult to see what happens to the other two terms – the 3x and the . These two terms are not what we call like terms because the x variables are raised to different powers. It would be like adding apples and oranges. We can add 2x + 3x and get 5x, and we can add to get , but we cannot add because they are not like terms. So if we add the expressions and , after simplifying we would have . All of the like terms have been added together.

Example

Simplify this expression:

Solution

First we decide which terms can be added together naturally and group them together like this:

Remember to keep track of which terms are positive and which are negative. The sign in front of each term must remain the same when we regroup. Note that we wrote the terms in descending order with regard to their exponents. Writing them in this order is standard practice in mathematics. Now all we need to do is add the like terms that we grouped in the parentheses to get:

Written without parentheses the simplified expression is:

Note that adding a negative is the same as subtracting, so that instead of adding (-5x), we subtracted 5x.