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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
·
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. |
