# Rational Expressions

When you want to add or subtract fractions, you need a
common denominator.

Of course, it makes life easier if you have the ** least common **

**denominator** of the expression. However, it is not completely necessary to

find the least commond denominator because you can simplify the expression

after you perform the operation. It just may be easier to find the least

common denominator at the beginning so you do not have to worry about

simplifying a complicated expression at the end. For example, say we want

to subtract

We can see that 4 = 2 · 2, so the least common denominator
is 4. Then, we

can solve this :

However, we can add general fractions by forcing the
denominators to be

common denominators (which may not be the least common denominator).

This is accomplished by multiplying both terms by 1, except we use a specific

way of writing it. We multiply by 1 as where
x is the denominator of the

**other** term:

Sometimes this is called "cross multiplying" because we
end up with a numerator

of the sum of the terms you would get if you made an X between the

fractions and multiplied the terms opposite each other, and the denominator

is the product of the denominators :

What we have here is ad from the \ part of the X, and bc
from the / part of

the X. Then, the denominator is the product of the old denominators.

We can easily write any subtraction as an addition problem as:

So, we now know how to add, subtract, multiply and divide
fractions .

Thankfully, rational expressions are nothing other than fractions. The

bad thing is that we do not know the numbers that are the numerator and

denominator because they depend on some variable. However, all we have to

worry about is not dividing by zero. For this reason, we have to make sure

that we take note of the x values that will make the denominator of** any**
of

the terms in the original expression zero. Then, we say the expression is not

defined for these x values by specifying our domain, as stated earlier. If you

know how to add, subtract, multiply and divide fractions, you know how to

add, subtract, multiply and divide rational expressions.

However, the problem comes in when we are faced with the task of reducing

the fraction. It is a **good idea** to factor the original expressions

completely before trying to perform any operations. When multiplying (and

dividing, since it is essentially a multiplication), it is a good idea not to
multiply

the terms out. After an addition or subtraction, it is a good idea to try

to factor again . Doing these makes it easier to:

• find the domain of the expression

• find the least common denominator of an expression

• simplify the expression by cancelling terms

Now that we have an idea of how to attack these problems, let's do some

examples.

First, we factor all of the expressions:

Now, we need to figure out the domain. From the first
term, we know we

can not have x + 2 = 0 i.e. x ≠ -2. In the second term, we can not have

a denominator of zero, so x ≠ 0, x ≠ 3 and x ≠ -3. Finally, since we are

dividing by the entire second term, it can not equal zero. Therefore, the

numerator of the second term can not be zero either. This happens when

x = 3 or x = 1. Putting it all together, our domain is { x ∈ R such that

x ≠ -3 and x ≠ -2 and x ≠ 0 and x ≠ 1 and x ≠ 3}. Next, we simplify:

We still have not done anything except find the domain and
simplify. Now,

we can " flip the second and multiply."

At this point, we try to simplify. However, there are no
common factors in

the numerator and denominator, so we can not do anything else. We can

either leave this as it is, or multiply it out. We will leave this as it is.

Now, consider a more complicated example.

This problem incorporates all of the operations of
fractions . However, there

is not much we can do to this. Order of operations tells us to do parenthases

then exponents . Niether of those appear in this equation. Then, we are supposed

to multiply and divide next. However, we **do not **know the reciprocal

of the entire denominator. Therefore, we have to simplify it. We can work

with the numerator and the denominator and try to make each of them one

fraction by performing the addition or subtraction:

Since we have not cancelled any terms, we have not changed
the domain

yet. So, we will get the domain from this expression. We know that xy^{2}

can not equal zero, and xy can not equal zero. Finally, since the entire big

denominator can not be zero, x^{2} -y^{2} can not equal zero. We
know from the

first two conditions that x ≠ 0 and y ≠ 0. Now, lets find when x^{2} - y^{2}
= 0.

This happens when x^{2} = y^{2}, or when x = ± y (because (-a)^{2}
= a^{2} for all

real numbers a). Therefore, our domain is { x ∈ R such that x ≠ 0, y ≠ 0

and x ≠± y}.

The expression can now be rewritten as

Now, we can factor and then multiply:

Now, we can simplify this by canceling xy:

At this point, there is no further simplification
possible. We can not factor

the numerator any more, so we will not be able to cancel with any of the

terms in the denominator.

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