Combining Like Terms

Like terms are those which have the same exponent AND the same variable. If an expression or equation has like terms, we can simplify it by combining these terms.

For example, in the expression $5x^2-x-2x^2$ there are two like terms – the $5x^2$ and the $-2x^2$ . Both terms have the variable x and an exponent of 2 so they can be combined. Let’s see how this works with the example:

$5x^2-x-2x^2$ $= 5x^2-2x^2-x$ (moving the like terms together)

$=3x^2-x$ (since 5-2=3)

The expression $3x^2-x$ is completely simplified since all of the like terms have been combined. The following are some examples of like and not-like terms:

$5x$ and $x^5$ . These terms are NOT like terms. They both have the same variable, but different exponents.
$4y^2$ and $-y^2$ . These terms ARE like terms since they both have the variable $y$ to the 2nd power (note that with the term $-y^2$ there is an “understood 1″. This is actually shorthand for $-1y^2$ )
$5xy$ and $2yx$ . These terms ARE like terms, but this is a bit different than what we have seen so far! Even when there is more than one variable, the same rule applies – there must be the same varible with the same exponent even if they are in a different order.
$5x^2y^3$ and $2x^2y^2$ . These terms are NOT like terms. While they both share the same variables and both of the x’s are squared, the y has a different exponent in each term.

Examples of Combining Like Terms

In each example, the idea is to simplify completely, that is until all like terms have been combined.

Example 1 Simplify $5x+2x-12x$ .

Solution:

$5x+2x-12x$

= $-5x$ . (all of the terms are like terms, so simply find 5+2-12)

Example 2 Simplify $5y^2-2y+y^2+3y-15$ .

$5y^2-2y+y^2+3y-15$ $=5y^2+y^2-2y+3y-15$ $=6y^2+y-15$ (remember, there is an understood 1 in front of the $y^2$ when there is no number written in front of the variable)

Example 3 Simplify $-2w^3+w^3-5w^2-3+w-6+3w^3$ .