In this section we concentrate on finding sums and
differences of rational expressions.
However, before we begin, we need to review some fundamental ideas and
technique.
First and foremost is the concept of the multiple of an
integer. This is best explained
with a simple example. The multiples of 8 is the set of integers {8k : k is an
integer}.
In other words, if you multiply 8 by 0, ±1, ±2, ±3, ±4, etc., you produce what
is
known as the multiples of 8.
Multiples of 8 are: 0, ±8, ±16, ±24, ±32, etc.
However, for our purposes, only the positive multiples are
of interest. So we will say:
Multiples of 8 are: 8, 16, 24 , 32, 40, 48 , 56, 64, 72 , .
. .
Similarly, we can list the positive multiples of 6.
Multiples of 6 are: 6, 12, 18, 24 , 30, 36, 42, 48 , 54, 60, 66,
72 , . . .
We’ve framed those numbers that are multiples of both 8
and 6. These are called the
common multiples of 8 and 6.
Common multiples of 8 and 6 are: 24, 48, 72, . . .
The smallest of this list of common multiples of 8 and 6
is called the least common
multiple of 8 and 6. We will use the following notation to represent the least
common
multiple of 8 and 6: LCM(8, 6).
Hopefully, you will now feel comfortable with the
following definition.
Definition 1. Let a and b be integers. The
least common multiple of a
and b, denoted LCM(a, b), is the smallest positive multiple that a and b
have in
common. |
For larger numbers, listing multiples until you find one
in common can be impractical
and time consuming. Let’s find the least common multiple of 8 and 6 a second
time, only this time let’s use a different technique .
First, write each number as a product of primes in
exponential form .
8 = 23
6 = 2 · 3
Here’ s the rule .
A Procedure to Find the LCM. To find the LCM of two integers , proceed as
follows.1. Express the prime equation s/what-types-of-factoring-for.html">factorization of each integer in exponential format.
2. To find the least common multiple, write down every prime number that
appears,
then affix the largest exponent of that prime that appears. |
In our example, the primes that occur are 2 and 3. The highest power of 2 that
occurs is 23. The highest power of 3 that occurs is 31. Thus, the LCM(8, 6) is
LCM(8, 6) = 23 · 31 = 24.
Note that this result is identical to the result found above by listing all
common multiples
and choosing the smallest.
Let’s try a harder example.
→ Example 2. Find the least common multiple of 24 and 36.
Using the first technique, we list the multiples of each number, framing the
multiples
in common.
Multiples of 24: 24, 48, 72 , 96, 120, 144 , 168, . . .
Multiples of 36: 36, 72 , 108, 144 , 180 . . .
The multiples in common are 72, 144, etc., and the least common multiple is
LCM(24, 36) =
72.
Now, let’s use our second technique to find the least common multiple (LCM).
First,
express each number as a product of primes in exponential format.
24 = 23 · 3
36 = 22 · 32
To find the least common multiple, write down every prime that occurs and affix
the
highest power of that prime that occurs. Thus, the highest power of 2 that
occurs is
23, and the highest power of 3 that occurs is 32. Thus, the least common
multiple is
LCM(24, 36) = 23 · 32 = 8 · 9 = 72.
Addition and Subtraction Defined
Imagine a pizza that has been cut into 12 equal slices. Then, each slice of
pizza
represents 1/12 of the entire pizza.
If Jimmy eats 3 slices, then he has consumed 3/12 of the entire pizza. If
Margaret
eats 2 slices, then she has consumed 2/12 of the entire pizza. It’s clear that
together
they have consumed
of the pizza. It would seem that adding two fractions with a common denominator
is
as simple as eating pizza! Hopefully, the following definition will seem
reasonable.
| Definition 3. To add two fractions with a common denominator, such as a/c
and b/c, add the numerators and divide by the common denominator. In symbols,
|
Note how this definition agrees precisely with our pizza consumption discussed
above. Here are some examples of adding fractions having common denominators.
Subtraction works in much the same way as does addition.
Definition 4. To subtract two fractions with a common denominator, such as
a/c and b/c, subtract the numerators and divide by the common denominator. In
symbols,
 |
Here are some examples of subtracting fractions already having common
denominators.
In the example on the right, note that it is extremely important to use grouping
symbols
when subtracting numerators. Note that the minus sign in front of the
parenthetical
expression changes the sign of each term inside the parentheses.
There are times when a sign change will provide a common denominator.
→ Example 5. Simplify
(6)
State all restrictions.
At first glance, it appears that we do not have a common denominator. On second
glance, if we make a sign change on the second fraction, it might help. So, on
the
second fraction, let’s negate the denominator and fraction bar to obtain
The denominators x−3 or 3−x are zero when x = 3. Hence, 3 is a restricted value.
For all other values of x, the left-hand side of
(7)
is identical to the right-hand side.
This is easily tested using the table utility on the graphing calculator, as
shown in
the sequence of screenshots in Figure 1. First load the left- and right-hand
sides of
equation (7) into Y1 and Y2 in the Y= menu of your graphing calculator, as shown
in
Figure 1(a). Press 2nd TBLSET and make the changes shown in Figure 1(b). Press
2nd TABLE to produce the table shown in Figure 1(c). Note the ERR (error)
message
at the restriction x = 3, but note also the agreement of Y1 and Y2 for all other
values
of x.
Figure 1. Using the table feature of the graphing calculator to check the result
in
equation (7).
