3.2 - Addition and Subtraction of Expressions

Before studying this topic you may wish to review the section on addition and subtraction of numbers.

Suppose that A and B are any two expressions. Adding B to A means setting up the sum as
(A) + (B),
and then simplifying as much as possible. Subtracting B from A means setting up the difference as
(A) − (B),
and then simplifying as much as possible.

The reason that we put brackets around A and B is that they are expressions, not just numbers, and the addition or subtraction is supposed to apply to whatever A and B may contain. (We want the addition or subtraction to be at the end in the order of operations.)

The simplifications that must be done are:

Example: If A = 4 x y + 7 and B = 8 y + 9 x y then setting up the sum A + B gives:
(4 x y + 7) + (8 y + 9 x y)
and setting up the difference A − B gives:
(4 x y + 7) − (8 y + 9 x y)

Removing the brackets

Removing the brackets from (A) + (B) is easy:

(A) + (B) always simplifies directly to A + B.     The brackets can just be dropped.

Here’s why: The acronym BEDMAS (Brackets, Exponents, Division and Multiplication, Addition and Subtraction) helps us remember the order of operations within an expression. Brackets are at the top of the priority list and addition and subtraction are at the bottom.

In the expression A + B the addition is already at the bottom of the list so we don’t need brackets around A and B to move them any higher in the priority list.

The same argument applies to A − B except in two cases. This is because in these two cases the subtraction is actually a multiplication by a negative, which is not at the bottom of the priority list! Here is the rule:

(A) − (B) simplifies to A − B except in these two cases:

Exception # 1:   If B is a negative monomial then the subtraction of a negative must be replaced by an addition.

Exception # 2:   If B is a multinomial then the − sign of the subtraction must be distributed over each term of the multinomial. The result, as explained below, is that each term of B has its sign reversed when the brackets are dropped.


Explanation of Exception 2: Why does distributing a − sign cause each term in the brackets to have its sign reversed? For example, why does a − (bc) simplify to ab + c ? Here is the sequence of steps with an explanation of each step:

a − (bc) ←    start with this
a − (b + (− c) ) change subtraction into addition of a negative
a + (− 1) · (b + (− c) ) change the other subtraction into multiplication by − 1
a + (− 1) · b + (− 1) · (− c)     use the distributive law to distribute the − 1
a  b + (− 1) · (− c)     change multiplication by − 1 back to subtraction
ab + c a negative times a negative gives a positive

Combining like terms

Recall that like terms are terms in an expression that differ only in their coefficients; they have identical variable factors. Like terms can be added or subtracted by adding or subtracting their coefficients (essentially by factoring with the distributive law).

Here are some examples. The first two examples are done in detail. Assume that x and y are variables. In the last example assume that a and b are constants. The like terms are shown in red.

Example: Simplify the expression

use the commutative law to sort the terms
use the distributive law to combine (factor) the two like terms

Example: Simplify the expression  

Example: Simplify the expression  

Example: Simplify the expression  
Note: In this last example the two terms in red were considered to be like terms because we were told that a and b are constants. If instead a and b were variables then the two terms in red would not be like terms and would NOT be combined.

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