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“Simplify” is a common instruction when doing algebra. It means to write what you’ve
got in a “simpler” way. That might mean with less terms, or less factors, or by reducing
fractions to simplest terms. (Yes, algebraic fractions can also be written in simplest
form.)
One of the most common things you’ll have to do to simplify an algebraic expression
is to collect like terms. “Like terms” are terms with the same variables, and the
same power (or number) of each variable in the term. For example, three f, f, seven f,
negative two f, and zero point four f are all “like terms”, because they all have
one f in the term. In contrast, none of those are “like” any of these. Eight e has an
e instead of an f. f squared has two fs (it’s f times f). Three f b has an extra variable.
And twenty-seven doesn’t have any fs.
Two y z, eight z y, and negative z y, are all “like terms”. The order of the variables
doesn’t matter: three times five is the same as five times three, it doesn’t matter
in what order you do the multiplication. So two y z and 8 z y are alike; they both have
one y and one z. By convention, we normally write them in alphabetic order: y z, rather
than z y, but it doesn’t really matter. And negative z y is just negative one z y;
the term has one y and one z in it. But these are not like them. y has no z at all. Eight
z y y has too many ys. y z squared means y, times z-squared (remember the order of operations
says that the exponent applies first before the times), so y z squared has one y but two
zs.
All of these have two as and three bs. Count them carefully. Remember the order they’re
written within the term doesn’t matter. Again, by convention we would normally write
them like the first one, in alphabetic order and using power notation: a squared, b cubed.
These are all not like those. Can you see why not?
To simply an expression that involves addition or subtraction, what you do is collect like
terms. Here’s a simple example. Three f and seven f are like terms: I have three f,
and another seven f, so altogether I have ten f.
Another example: two y z, minus z y, plus y. Collect like terms. Can you see any like
terms? y z and z y are like terms, because although the order of the variables is different,
they have the same number of each variable. What’s the coefficient of the second one?
It’s one. So two y z minus z y is just one y z. And the y has no other like terms, so
we have to just leave it alone: y z plus y. See? It’s simpler than what I started with.
Five e minus seven b plus three e plus two b. Now be careful here. Minus signs go with
the term immediately afterwards. This minus sign goes with the seven.
Can you see any like terms? There are es, and bs. So put them together. And be careful
with that minus sign. It needs to go with the seven. Don’t leave it behind. The minus
sign applies to the term straight after it, and nothing else.
Five e, plus three e, is eight e. Minus seven b, plus two b, is minus five b. Careful with
that minus sign. If you’re going to do the minus and the plus in the order they’re
written, you have to go “minus seven, plus two”. You can’t say “minus, seven plus
two”, for minus nine, because that would be doing the plus before the minus. Keep the
minus sign with the term straight after it, and nothing else.
Let’s try a couple of harder ones.
Four p q squared, plus two q p squared, minus eight q squared, plus three q squared p. Okay,
can you see any like terms? Yes, the p q squared and the q squared p. They’re like terms
because they both have one p and two qs. But the two q p squared is not like those. It
has one q and two ps. And the eight q squared has two qs but no p.
So collect the like terms, four p q squared and three q squared p. Put them together.
The others can stay at the end. Notice I’ve written them all in the more conventional
alphabetic order. Be careful with that minus sign: keep it with the eight q squared that
follows it. The minus sign applies only to the term immediately after it.
Okay, now four plus three is seven p q squared. And the rest is unchanged.
One more. Three y y n, minus two y n squared, minus seven y n y, plus n y y, plus eight
n n y. Now remember, the order of the multiplied variables doesn’t matter. So let’s write
them all conventionally first, in alphabetic order with power notation. y y is y squared,
and put the n first. Write the n first. Put the n first, and then y times y is y squared.
y squared again. n squared.
Okay, now is it easier to see the like terms? Collect them together. Careful: keep the minus
signs with the terms that follow them. You can also write a one in front of the third
term, if it makes it easier for you.
Now you can simplify. Three minus seven is negative four, plus one is negative three,
n y squared. Negative two plus eight is positive six n squared y. And that’s much simpler
than the first line. Good.