12.1 Simplifying Rational Expressions
A rational expression is an expression that can be written in the form
P
where P and Q are both
Q
polynomials and Q ≠ 0.
Examples of Rational Expressions:
Evaluating Rational Expressions
To evaluate a rational expression for a particular value(s), substitute the replacement value(s) into the
rational expression and simplify the result.
Examples: Evaluate.
1. Evaluate the following expression for
−2
−5 +
= −2.
2. Evaluate the expression for
2 +5
5 − 15
= −3.
Undefined Rational Expressions
To find values for which a rational expression is undefined, find values for which the denominator is 0.
Examples: Find any values that make the following rational expression undefined.
3.
4.
5.
6.
Simplifying Rational Expressions
Simplifying a rational expression means writing it in lowest terms or simplest form. To do this, we need
to use the Fundamental Principle of Rational Expressions
If ,
and
are polynomials, and
To Simplify a Rational Expression
and
are not 0, then
= .
Step 1: Completely factor the numerator and denominator.
Step 2: Divide out factors common to the numerator and denominator. (This is the same as
“removing a factor of 1.”)
When simplifying a rational expression, we look for common factors, not common terms.
Examples: Simplify.
7.
8.
10.
11.
9.
(
(
)(
)(
)
)
12.
12.2 Multiplying and Dividing Rational Expressions
Multiplying Rational Expressions
If
and
are rational expressions, then
∙ =
.
To multiply rational expressions, multiply the numerators and then multiply the denominators.
After multiplying such expressions, simplify if possible.
To Multiply Rational Expressions
Step 1: Completely factor numerators and denominators.
Step 2: Multiply numerators and multiply denominators.
Step 3: Simplify or write the product in lowest terms by dividing out common factors.
Examples: Multiply and simplify, if possible.
1.
3.
∙
2.
∙
4.
∙
∙
Dividing Rational Expressions
If
and
are rational expressions, then ÷ =
∙ =
.
To divide two rational expressions, multiply the first rational expression by the reciprocal of the second
rational expression.
Remember, to Divide by a Rational Expression, multiply by its reciprocal.
Examples: Divide and simplify, if possible.
5.
7.
÷
6.
÷
8.
(
)
÷
÷
12.3 Adding and Subtracting Rational Expressions with Same Denominators and
Lowest Common Denominator (LCD)
Adding and Subtracting Rational Expressions with Common Denominators
If
and
are rational expressions, then
+
=
and
−
=
.
To add or subtract rational expressions, add or subtract numerators and place the sum or difference over
the common denominator.
Examples: Add or subtract as indicated.
1.
3.
+
−
2.
4.
+
−
Least Common Denominator (LCD)
To add or subtract rational expressions with different denominators, you have to change them to
equivalent forms that have the same denominator (a common denominator).
This involves finding the least common denominator of the two original rational expressions.
To Find the Least Common Denominator
Step 1:
Factor each denominator completely.
Step 2:
The least common denominator (LCD) is the product of all
unique factors found in Step 1, each raised to a power equal to
the greatest number of times that the factor appears in any one
factored denominator.
Examples: Find the LCD.
5.
7.
,
6.
,
8.
,
,
,
9.
,
10.
Writing Equivalent Rational Expressions
To change rational expressions into equivalent forms, we use the principal that multiplying by 1 (or any
form of 1), will give you an equivalent expression.
=
∙1=
∙
∙
∙
=
Examples: Write the Equivalent Rational Expression.
11.
13.
=
=
12.
=
(
)(
)
14.
(
(
)
)
=
12.4 Adding and Subtracting Rational Expressions with Different Denominators
As stated in the previous section, to add or subtract rational expressions with different denominators, we
have to change them to equivalent forms first.
To Add or Subtract Rational Expressions with Different Denominators
Step 1:
Find the LCD of all the rational expressions.
Step 2:
Rewrite each rational expression as an equivalent
expression whose denominator is the LCD found in Step 1.
Step 3:
Add or subtract numerators and write the sum or
difference over the common denominator.
Step 4:
Simplify or write the rational expression in lowest terms.
Examples: Add or Subtract.
1.
3.
+
2.
−3
4.
−
+
12.5 Solving Equations containing Rational Expressions
To solve equations containing rational expressions, clear the fractions by multiplying both sides of the
equation by the LCD of all the fractions. Then solve the simplified equation as in previous sections.
Note: Find the values that would make the problem undefined.
Examples: Solve.
1.
+
3. 2 −
+4=9
=
2.
=8
4. 6 + =
+7
5.
7.
=
+
6.
+1=
8.
3+
=
=
9.
11.
+ =
+1=
10.
+ =
12.
−
=
12.6 Solving Equations containing Rational Expressions
Examples: Solve.
1. Sally can paint a room in 9 hours while it takes
Steve 7 hours to paint the same room. How long
would it take them to paint the room if they
worked together?
2. In 4 minutes, a conveyor belt moves 300
pounds of recyclable aluminum cans from a truck
to a storage area. A smaller belt moves the same
quantity of aluminum cans the same distance in 9
minutes. If both conveyor belts are used, find how
long it takes to move the cans to the storage area.
3. Marcus and Tony work for the same company.
Marcus lays a slab of concrete in 2 hours. Tony lays
the same size slab in 7 hours. If both work on the
job together and the cost of labor is $45.00 per
hour, decide what the labor estimate should be.
4. One pipe fills a storage pool in 7 hours. A
second pipe fills the same pool in 21 hours. When a
third pipe is added and all three are used to fill the
pool, it takes only 3 hours. Find how long it takes
the third pipe to do the job alone.
5. In 7 hours, an experienced pastry chef
prepares enough pies to supply a local restaurant’s
daily order. Another pastry chef prepares the same
number of pies in 9 hours. Together with a third
pastry chef, they prepare the pies in 2 hours. Find
how long it takes the third pastry chef to prepare
the pies alone.
6. One pipe fills a storage pool in 7 hours. A
second pipe fills the same pool in 21 hours. When a
third pipe is added and all three are used to fill the
pool, it takes only 3 hours. Find how long it takes
the third pipe to do the job alone.