Download Simplifying Rational (Fractional) Expressions To simplify an

Simplifying Rational (Fractional)
Expressions
To simplify an algebraic fraction means to reduce it to lowest terms.
This is done by dividing out the common factors in the numerator and the denominator.
Example 1
Simplify:
First, factor the numerator and factor
the denominator.
Then divide out the common binomial
factor (x + 2).
Final Answer:
Example 2
Simplify:
Again, factor the numerator and
factor the denominator.
Divide out the common binomial
factor of (y - 2)
Divide out the common monomial
factor of 2y
Final Answer:
Example 3
Simplify:
Factor the numerator and
factor the denominator.
Notice how factoring (-1) will allow
for the division of the common binomial
factor of (x - 2).
Final Answer:
or
Adding and Subtracting Rational
(Fractional) Expressions
Topic Index | Algebra2/Trig Index | Regents Exam Prep Center
OKAY! ... so Algebraic Fractions are
challenging. But you CAN DO THESE
problems! The secret is to work slowly
and show all of your work.
Fractions are fractions!
It doesn't matter if the fractions are made
up of numbers, or of algebraic variables,
the rules are the same!
Examine the basic process:
Add:
You CANNOT reduce
(cancel) those 3's!!!!
The Basic RULE for Adding and
Subtracting Fractions:
Get a Common
Denominator!
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.
Note: The smallest common denominator is
called the "least common denominator" or
LCD.
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:
1.
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).
2.
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).
3.
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.
4.
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.
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.
5.
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).
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.
Factoring is your friend!
(Factoring lets you find the
smallest common
denominator quickly, thus
making your work easier.)
6.
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!!!
Multiplication of Rational
(Fractional) Expressions
Simply Put:
The rule for multiplying algebraic fractions
is the same as the rule for multiplying numerical fractions.
Multiply the tops (numerators)
AND
multiply the bottoms (denominators).
The Rule:
If possible, reduce (cancel) BEFORE you multiply the
tops and bottoms!
(It's easier than simplifying at the end!)
EXAMPLES
Multiply and express the product in simplest form:
1.
CAUTION:
You can only cancel
top with bottom or
bottom with top.
There is NO canceling
bottom with bottom or
top with top!
2.
Remember to factor
first if possible.
3.
Factor everything
that can be factored.
Cancel (bottom with
top or top with
bottom).
Multiply tops
(numerators) and
bottoms
(denominators).
Division of Rational (Fractional)
Expressions
Simply Put:
The rule for dividing algebraic fractions
is the same as the rule for dividing numerical fractions.
Change the division sign to multiplication,
flip the 2nd fraction ONLY,
and then follow the steps for "multiplying rational expressions".
The
Rule:
EXAMPLES
Divide and express the quotient in simplest form:
1.
Change division to
multiplication and
flip the 2nd fraction.
Now follow the
steps for
Multiplying
Rational Fractions.
2.
Change division to
multiplication and
flip the 2nd fraction.
Now follow the
steps for
Multiplying
Rational Fractions.