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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.
Examples:
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
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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
Minus Signs: Subtraction and Negatives
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.
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