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Combining ("Gathering") Like Terms

First, we'll define what "terms" are. Then we'll see what "like terms" are.

A term is a product of a number and some variables, like $3 x y$ or $- 4 x^{2}$ . If the variable part is the same in two terms, they're called like terms .

Examples :

$3 x$ and $5 x$ are like terms; $3 x$ and $5 y$ are unlike terms.

$7 a^{3} b$ and $99 a^{3} b$ are like terms; $7 a b$ and $99 a b^{2}$ are unlike.

If two terms are "like" then you can add or subtract them. Notice the use of the Distributive Property here.

$3 x + 5 x = (3 + 5) x = 8 x$

The variable part stays the same; we can just add the coefficients (the numbers in front of the variables).

But unlike terms can't be added.

$3 x + 4 y \neq 7 x$ or $7 y$ or $7 x y$ , and

$6 + 5 x \neq 11 x$ .

(The symbol $\neq$ means "not equal to.")

This is important when simplifying polynomials .

Example:

Simplify.

$6 x^{2} + 5 x + 4 - 4 x^{2} + 7 x - 8$

First, collect the like terms in parentheses.

$= (6 x^{2} - 4 x^{2}) + (5 x + 7 x) + (4 - 8)$

Then simplify.

$= 2 x^{2} + 12 x - 4$