Terms and Factors

A term in an algebraic expression is an expression involving letters and/or numbers (called factors), multiplied together.

Example 1

The algebraic expression

5x

is an example of one single term. It has factors 5 and x.
The 5 is called the coefficient of the term and the x is a variable.

Example 2

5x + 3y has two terms.

First term: 5x, has factors 5 and x
Second term: 3y, has factors 3 and y

The 5 and 3 are called the coefficients of the terms.

Example 3

The expression

3x² − 7ab + 2e√π

has three terms.

First term: 3x²; has factors 3 and x²
Second term: -7ab; has factors -7, a and b
Third Term: 2e√π; has factors 2, e and √π

The 3, -7 and 2 are called coefficients of the terms.

Like Terms

"Like terms" are terms that contain the same variables raised to the same power.

Example 4

3x² and 7x² are like terms.

Example 5

-8x² and 5y² are not like terms, because the variable is not the same.

Adding and Subtracting Terms

Important: We can only add or subtract like terms.
Why? Think of it like this. On a table we have 4 pencils and 2 books. We cannot add the 4 pencils to the 2 books - they are not the same kind of object.
We go get another 3 pencils and 6 books. Altogether we now have 7 pencils and 8 books. We can't combine these quantities, since they are different types of objects.
Next, our sister comes in and grabs 5 pencils. We are left with 2 pencils and we still have the 8 books.
Similarly with algebra, we can only add (or subtract) similar "objects", or those with the same letter raised to the same power.

Example 6

Simplify 13x + 7y − 2x + 6a

Answer

13x + 7y − 2x + 6a
The only like terms in this expression are 13x and -2x. We cannot do anything with the 7y or 6a.
So we group together the terms we can subtract, and just leave the rest:

(13x − 2x) + 6a + 7y
= 6a + 11x + 7y

We usually present our variables in alphabetical order, but it is not essential.

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Example 7

Simplify −5[−2(m − 3n) + 4n]

Answer

Go back to Order of Operations if you are not sure what to do first with this question.

−5[−2(m − 3n) + 4n]

The square brackets [ ] work just the same as round brackets ( ). We could have used curly brackets { } here as well.
The first thing we do is expand out the round brackets inside.

−2(m − 3n) = −2m + 6n

The negative times negative in the middle gives positive 6n.
Now add the 4n in the square brackets:

[−2m + 6n + 4n] = [−2m + 10n]

Remembering the −5 out front, our problem has become:

−5[−2m + 10n] = 10m − 50n

Taking each term one at a time, what we did was:
−5 × −2m = 10m (Two negative numbers multiplied together give a positive); and
−5 × 10n = −50n (Negative times positive gives negative)
Go back to the section on Integers if you are not sure about multiplying with negative numbers.
So here's the answer:

−5[−2(m − 3n) + 4n] = 10m − 50n

Note:
The fancy name for round brackets ( ) is "parentheses".
The fancy name for square brackets [ ] is "box brackets".
The fancy name for curly brackets { } is "braces".

Example 8

Simplify −[7(a − 2b) − 4b]

Answer

−[7(a − 2b) − 4b]
= −[7a − 14b − 4b]
= −[7a − 18b]
= −7a + 18b