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OKAY! ... so Algebraic
Fractions are challenging. But you CAN DO THESE
problems! The secret is to work slowly and show all of
your work. |
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Fractions are fractions!
It doesn't matter if the fractions are made
up of numbers,
or of algebraic variables,
the rules are the same! |
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The Basic RULE for Adding and Subtracting
Fractions:
Get a Common
Denominator! |
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Examine the basic process:
Add:
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Get a common
denominator - the smallest number that both denominators can divide
into
without remainders. In this case, the number is 12.
To change the denominator of 3 into 12 requires multiplying by
4. To change the denominator of 4 into 12 requires multiplying
by 3.
With each fraction, whatever is multiplied times the bottom must ALSO be
multiplied times the top.
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You CANNOT
reduce (cancel)
those 3's!!!! |
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Note: The
smallest common denominator is called the "least common
denominator" or LCD. |
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Do not add the common
denominators. Add only the numerators (tops). |
Multiplying the top and bottom by the same number is multiplying by
the multiplicative identity element (which is = 1) and therefore does not change the
fraction.
Now, apply this basic process
to algebraic fractions:
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1. |
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The
Least Common Denominator (LCD) is 8.
Multiply BOTH the top and bottom of the first fraction by 2 to create the common
denominator.
Remember, when adding "like" variables, add only
the numbers in front of the variables (the coefficients). |
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2. |
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The
Least Common Denominator is 12x.
If the denominator contains
monomials
with variable(s), the common denominator will need to contain the
variable(s) with the largest power(s).
Since the largest power of x in this problem is 1, the common
denominator must contain an x to the first power, multiplied
by a common denominator for the 6, 3 and 4 (which is 12).
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| 3. |
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The
Least Common Denominator is
 . The largest power needed for
both a and
b in this problem is a power of 2.
Remember, when
working with
binomials, like
(a - b) and (a + b), think
of each binomial as ONE entity. For example, in (a - b), the
a cannot
go anywhere without his buddy (-b)
tagging along. You can never reduce (cancel) only the
a or only the b
in (a - b).
Notice that when we add in
this problem, the ab terms are eliminated.
The final answer cannot be reduced further.
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| 4. |
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The
Least Common Denominator is a•(a - 5).
Notice that one of the denominators now contains a
binomial,
(a - 5).
Remember that the
a
in (a - 5) cannot
go anywhere without his buddy (-5)
tagging along. You cannot separate the
a from the -5,
as they must come as a pair. In addition, you can never reduce
(cancel) only the
a or only the
5
in
(a - 5).
The final answer may
be expressed in several ways.
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5. |
Be
careful when combining fractions under subtraction!
Be sure to subtract the ENTIRE numerator value behind the
subtraction sign. In this problem, when "subtracting" (x - 6) it
is necessary to distribute the
negative (subtraction) sign across the parentheses, creating
-x + 6. |
The Least
Common Denominator is
x(x - 6)(x + 6).
When working with polynomial denominators, always consider the possibility of "factoring" to find the LCD.Instead
of just multiplying the two denominators together to find the common
denominator, first look to see if it
is possible to find the least common denominator (LCD) by
factoring and
looking at those results. Factoring may lead to a simpler and
faster solution for the LCD.
Both of these
denominators can be factored.
and
Wow! They were each hiding a factor of (x + 6).
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Factoring is
your friend! |
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(Factoring lets you find the
smallest common denominator quickly, thus making your work easier.) |
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6. |
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Tricky one!!!
At first glance it appears that we will
need to multiply both denominators to get the common denominator - but look more
closely!
If you multiply one of the denominators by
(-1), you will create a factor that matches the other denominator.
2x - 6 = (-1)(6 - 2x) = (-6 + 2x) = 2x - 6
(one denominator is the additive inverse of the other)
When
multiplying one of the denominators by
(-1), be sure to multiply its numerator by
(-1) also! This technique can
also be accomplished by "factoring out a -1" from one of the
denominators. |
Whew!!!
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