Basic Math and Pre-Algebra For Dummies (96 page)

 When a term appears to have no coefficient, the coefficient is actually 1. So the coefficient of
x
²
is 1, and the coefficient of –
x
is –1. And when a term is a constant (just a number), that number with its associated sign is the coefficient. So the coefficient of the term –7 is simply –7.

By the way, when the coefficient of any algebraic term is 0, the expression equals 0 no matter what the variable part looks like:

• 0 x = 0

  0 xyz = 0

  0 x 3 y 4 z 10 = 0

In contrast, the
variable part
of an expression is everything except the coefficient. The previous table shows the four terms of the same expression, with each term's variable part.

Like terms
(or
similar terms
) are any two algebraic terms that have the same variable part — that is, both the letters and their exponents have to be exact matches. Here are some examples:

As you can see, in each example, the variable part in all three like terms is the same. Only the coefficient changes, and it can be any real number: positive or negative, whole number, fraction, or decimal — or even an irrational number such as π. (For more on real numbers, see Chapter
25
.)

In this section, I get you up to speed on how to apply the Big Four to algebraic expressions. For now, just think of working with algebraic expressions as a set of tools that you're collecting, for use when you get on the job. You find how useful these tools are in Chapter
22
, when you begin solving algebraic equations.

 Add like terms by adding their coefficients and keeping the same variable part.

For example, suppose you have the expression 2
x
+ 3
x.
Remember that 2
x
is just shorthand for
x
+
x,
and 3
x
means simply
x
+
x
+
x.
So when you add them up, you get the following:

• =
  x
  +
  x
  +
  x
  +
  x
  +
  x
  = 5
  x
  

As you can see, when the variable parts of two terms are the same, you add these terms by adding their coefficients: 2
x
+ 3
x
= (2 + 3)
x
. The idea here is roughly similar to the idea that 2 apples + 3 apples = 5 apples.

 You
cannot
add nonlike terms. Here are some cases in which the variables or their exponents are different:

In these cases, you can't simplify the expression. You're faced with a situation that's similar to 2 apples + 3 oranges. Because the units (apples and oranges) are different, you can't combine terms. (See Chapter
4
for more on how to work with units.)

 Subtraction works much the same as addition. Subtract like terms by finding the difference between their coefficients and keeping the same variable part.

For example, suppose you have 3
x
–
x.
Recall that 3
x
is simply shorthand for
x
+
x
+
x.
So doing this subtraction gives you the following:

• x
  +
  x
  +
  x
  –
  x
  = 2
  x
  

No big surprises here. You simply find (3 – 1)
x.
This time, the idea roughly parallels the idea that $3 – $1 = $2.

Here's another example:

• 2
  x
  – 5
  x
  

Again, no problem, as long as you know how to work with negative numbers (see Chapter
4
if you need details). Just find the difference between the coefficients:

• = (2 – 5)
  x
  = –3
  x
  

In this case, recall that $2 – $5 = –$3 (that is, a debt of $3).

 You
cannot
subtract nonlike terms. For example, you can't subtract either of the following:

As with addition, you can't do subtraction with different variables. Think of this as trying to figure out $7 – 4 pesos. Because the units in this case (dollars versus pesos) are different, you're stuck. (See Chapter
4
for more on working with units.)

 Unlike adding and subtracting, you can multiply nonlike terms. Multiply
any
two terms by multiplying their coefficients and combining — that is, by collecting or gathering up — all the variables in each term into a single term, as I show you next.

For example, suppose you want to multiply 5
x
(3
y
). To get the coefficient, multiply 5 × 3. To get the algebraic part, combine the variables
x
and
y:

• = 5(3)
  xy
  = 15
  xy
  

Now suppose you want to multiply 2
x
(7
x
). Again, multiply the coefficients and collect the variables into a single term: