By “simplifying” an algebraic expression, we mean writing it in the most compact or efficient manner, without changing the value of the expression. This mainly involves collecting like terms, which means that we add together anything that can be added together. The rule here is that only like terms can be added together.
Like terms are those terms which contain the same powers of same variables. They can have different coefficients, but that is the only difference.
3x, x, and –2x are like terms.
2x2, –5x2, and are like terms.
xy2, 3y2 x, and 3xy2 are like terms.
xy2 and x2 y are NOT like terms, because the same variable is not raised to the same power.
Combining like terms is permitted because of the distributive law. For example,
3x2 + 5x2 = (3 + 5)x2 = 8x2
What happened here is that the distributive law was used in reverse—we “undistributed” a common factor of x2 from each term. The way to think about this operation is that if you have three x-squareds, and then you get five more x-squareds, you will then have eight x-squareds.
Example: x2 + 2x + 3x2 + 2 + 4x + 7
Starting with the highest power of x, we see that there are four x-squareds in all (1x2 + 3x2). Then we collect the first powers of x, and see that there are six of them (2x + 4x). The only thing left is the constants 2 + 7 = 9. Putting this all together we get
x2 + 2x + 3x2 + 2 + 4x + 7
= 4x2 + 6x + 9
· Parentheses must be multiplied out before collecting like terms
You cannot combine things in parentheses (or other grouping symbols) with things outside the parentheses. Think of parentheses as opaque—the stuff inside the parentheses can’t “see” the stuff outside the parentheses. If there is some factor multiplying the parentheses, then the only way to get rid of the parentheses is to multiply using the distributive law.
Example: 3x + 2(x – 4)
= 3x + 2x – 8
= 5x – 8
Subtraction can be replaced by adding the opposite
3x – 2 = 3x + (–2)
A special case is when a minus sign appears in front of parentheses. At first glance, it looks as though there is no factor multiplying the parentheses, and you may be tempted to just remove the parentheses. What you need to remember is that the minus sign indicating subtraction should always be thought of as adding the opposite. This means that you want to add the opposite of the entire thing inside the parentheses, and so you have to change the sign of each term in the parentheses. Another way of looking at it is to imagine an implied factor of one in front of the parentheses. Then the minus sign makes that factor into a negative one, which can be multiplied by the distributive law:
3x – (2 – x)
= 3x + (–1)[2 + (–x)]
= 3x + (–1)(2) + (–1)(–x)
= 3x – 2 + x
= 4x – 2
However, if there is only a plus sign in front of the parentheses, then you can simply erase the parentheses:
3x + (2 – x)
= 3x + 2 – x
Although you can always explicitly replace subtraction with adding the opposite, as in this previous example, it is often tedious and inconvenient to do so. Once you get used to thinking that way, it is no longer necessary to actually write it that way. It is helpful to always think of minus signs as being “stuck” to the term directly to their right. That way, as you rearrange terms, collect like terms, and clear parentheses, the “adding the opposite” business will be taken care of because the minus signs will go with whatever was to their right. If what is immediately to the right of a minus sign happens to be a parenthesis, and then the minus sign attacks every term inside the parentheses.