|
When we reduce a common fraction
such as
we do so by noticing that
there is a factor common to
both the numerator and the
denominator (a factor of 2
in this example), which we
can divide out of both the
numerator and the denominator.
We use exactly the same procedure
to reduce rational expressions.
Each term in the numerator
must have a factor that cancels
a common factor in the denominator.
 ,
but
cannot be reduced because
the 2 is not a common factor
of the entire numerator.
WARNING
You can only cancel
a factor of the entire
numerator with a factor of
the entire denominator
| However, as an alternative,
a fraction with more
than one term in the
numerator can be split
up into separate fractions
with each term over
the same denominator;
then each separate fraction
can be reduced if possible:
·
Think
of this as the reverse
of adding fractions
over a common denominator.
Sometimes this is a
useful thing to do,
depending on the circumstances.
You end up with simpler
fractions, but the price
you pay is that you
have more fractions
than you started with. |
·
Polynomials
must be factored first. You
can’t cancel factors unless
you can see the factors:
Example:
·
Notice how canceling
the (x – 2)
from the denominator left
behind a factor of 1
Same rules as for rational numbers!
Multiplication
- Both
the numerators and the denominators
multiply together
- Common
factors may be cancelled
before multiplying
Example:
|
Given
Equation: |
|
|
First
factor all the expressions:
(I also put the denominators
in parentheses because
then it is easier to
see them as distinct
factors) |
|
|
Now
cancel common factors—any
factor on the top can
cancel with any factor
on the bottom: |
|
|
Now
just multiply what’s
left.
You usually do not have
to multiply out the
factors, just leave
them as shown. |
|
Division
- Multiply
by the reciprocal of the
divisor
- Invert
the second fraction, then
proceed with multiplication
as above
- Do
not attempt to cancel factors
before it is written as
a multiplication
Same procedure as for rational numbers!
·
Only
the numerators can add together,
once all the denominators
are the same
Finding the LCD
- The
LCD is built up of all the
factors of the individual
denominators, each factor
included the most number
of times it appears in an
individual denominator.
- The product of all the denominators is always a common denominator,
but not necessarily the
LCD (the final answer may
have to be reduced).
Example:
|
Given
equation: |
|
|
Factor
both denominators: |
|
|
Assemble
the LCD:
Note that the LCD contains
both denominators |
|
|
Build
up the fractions so
that they both have
the LCD for a denominator:
(keep both denominators
in factored form to
make it easier to see
what factors they need
to look like the LCD) |
|
|
Now
that they are over the
same denominator, you
can add the numerators: |
|
|
And
simplify: |
|
|
