Introduction
Expressions come in a variety of types, including rational
expressions. A rational expression is a ratio of two
polynomials, in which a variable appears in the
3x
denominator; for example,
is a rational expression.
x +1
Working with rational expressions can often be made
easier by analyzing them to uncover more familiar (and
sometimes less complex) structures within them. In this
section, you will explore some of those structures.
1
3.1.1: Adding and Subtracting Rational Expressions
Introduction, continued
Where one rational expression exists, another may as
well. Extracting meaning from the context may require
these expressions to be combined in order to determine
a sum or difference. Combining rational expressions
through addition or subtraction is not complex, though it
does demand attention to detail. Rewriting equivalent
fractions can often simplify the solution process.
2
3.1.1: Adding and Subtracting Rational Expressions
Key Concepts
• Before adding or subtracting rational expressions, you
must find a common denominator. A common
denominator is a quantity that is a shared multiple of the
denominators of two or more fractions.
3
3.1.1: Adding and Subtracting Rational Expressions
Key Concepts, continued
• A common denominator can be determined by finding
the product of the denominators. For example, a
common denominator of the rational expression
3x
2x + 1
is found by multiplying the two
+ 2
x -1 x + x - 2
denominators, x – 1 and x2 + x – 2:
(x – 1)(x2 + x – 2) = x3 – 3x + 2.
4
3.1.1: Adding and Subtracting Rational Expressions
Key Concepts, continued
• Once a common denominator has been found, it can
be used to write equivalent rational expressions for
each term of the sum (or difference, if subtracting).
• Using common denominators, the same rational
3x
2x + 1
expression as before,
, can be
+ 2
x -1 x + x - 2
rewritten in an equivalent form, as shown on the next
slide.
5
3.1.1: Adding and Subtracting Rational Expressions
Key Concepts, continued
3x
2x + 1
+
x - 1 x2 + x - 2
=
=
3x
•
x2 + x - 2
x -1 x + x - 2
2
3x 3 + 3x 2 - 6x
x - 3x + 2
3
+
+
2x + 1
•
x -1
x + x - 2 x -1
2
2x 2 - x - 1
x3 - 3x + 2
6
3.1.1: Adding and Subtracting Rational Expressions
Key Concepts, continued
• Expressed with a common denominator, the sum of
rational expressions is the sum of the numerators:
x y x+y
. Thus, we can rewrite the expressions
+ =
z z
z
over a single denominator, as shown.
3x 3 + 3x 2 - 6x
x - 3x + 2
3
+
2x 2 - x - 1
x - 3x + 2
3
=
3x3 + 3x 2 - 6x + 2x 2 - x - 1
x3 - 3x + 2
7
3.1.1: Adding and Subtracting Rational Expressions
Key Concepts, continued
• Expressed with a common denominator, the difference
of rational expressions is the difference of the
x y x-y
numerators: - =
z z
z
8
3.1.1: Adding and Subtracting Rational Expressions
Key Concepts, continued
• The least common denominator (LCD) is the least
common multiple of the denominators of two or more
fractions. In other words, it’s the smallest possible
common denominator. The LCD can be determined by
finding the product of all the unique factors of the
denominator.
9
3.1.1: Adding and Subtracting Rational Expressions
Key Concepts, continued
• Our sample rational expression,
3x
+
2x + 1
x -1 x + x - 2
is equivalent to the rational expression
3x
2x + 1
. The LCD is (x – 1)(x + 2), or
+
x - 1 (x - 1)(x + 2)
x2 – x – 2.
2
,
• Recall that a rational expression cannot include a value
in the denominator that causes it to equal 0, since 0 is
undefined in the denominator.
10
3.1.1: Adding and Subtracting Rational Expressions
Common Errors/Misconceptions
• canceling individual terms of the numerator or
denominator, rather than factors
• distributing values incorrectly, especially negative
numbers
• forgetting to check for invalid values in the denominator
• neglecting to keep track of signs (particularly when
subtracting terms that must first be properly distributed)
11
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice
Example 3
Simplify the rational expression
5
x
2
+
3
x
+
1
4x
.
12
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 3, continued
1. Identify any invalid values of the
expression.
In the first rational term, x2 ≠ 0. Taking the square
root of each side results in x ≠ 0. This same value
also makes the other two denominators invalid. We
only need to avoid this one value, so the domain of
the expression is x ≠ 0.
13
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 3, continued
2. Find a common denominator.
As in previous examples, we can determine a
common denominator by finding the product of the
denominators. That gives us x2 • x • 4x = 4x4. This
would work just fine (it is most certainly a common
denominator), however, it does seem large. We can
likely determine a smaller common denominator.
14
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 3, continued
Each of the three denominators in the original
expression has x as a factor, so we only need to
include it once. The first term has an additional factor
of x, and the last term has an additional term of 4.
The product of these three values—the common x,
another x from the first term, and the 4 from the last
term—is 4x2. This is a smaller and perhaps easier to
handle denominator.
Thus, we can use 4x2 as the common denominator.
15
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 3, continued
3. Rewrite each term of the expression using
the new denominator.
5
x
2
+
3
x
+
1
Original expression
4x
æ 4 5 ö æ 4x 3 ö æ x 1 ö
= ç • 2÷ +ç • ÷ +ç • ÷
è 4 x ø è 4x x ø è x 4x ø
Multiply
5
by
4
4
x
3
4x
by
, and
x
4x
1
x
by .
4x
x
2
,
16
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 3, continued
=
20
4x
2
+
12x
4x
2
+
x
4x
2
Simplify each term.
Rewrite the expression
20 + 12x + x
by adding the
=
numerators
over the
4x 2
common denominator.
20 + 13x
=
Combine like terms.
4x 2
The rewritten expression is
20 + 13x
4x
3.1.1: Adding and Subtracting Rational Expressions
2
.
17
Guided Practice: Example 3, continued
4. Check to see if the result can be written in
a simpler form.
There are no common factors of the numerator or
denominator. This expression cannot be written in
any other format that will simplify the result.
20 + 13x
Therefore,
4x 2
, where x ≠ 0, is the simplest way to rewrite
the original expression.
18
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 3, continued
You can use this strategy to combine many different
rational expressions, including those that appear to be
more complex. The expressions may involve more
terms, or perhaps subtraction here and there. Yet the
process outlined in this example remains the same.
Carefully following these same steps will yield
successful results.
✔
19
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 3, continued
20
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice
Example 4
Simplify the rational expression
3x
+
8
2x + 1 x - 3
.
21
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 4, continued
1. Identify any invalid values of the
expression.
The first rational term has a denominator of 2x + 1.
Set the denominator equal to 0 and then solve the
resulting equation for x to determine values that make
this expression invalid.
2x + 1 = 0
2x = –1
x=-
1
2
3.1.1: Adding and Subtracting Rational Expressions
22
Guided Practice: Example 4, continued
1
An x-value of - invalidates the expression,
2
1
so x ¹ - .
2
The second rational term has a denominator of x – 3.
Again, set the denominator equal to 0 and then solve
the resulting equation for x to determine values that
make this expression invalid.
23
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 4, continued
x–3=0
x=3
An x-value of 3 invalidates the expression, so x ≠ 3.
3x
8
The rational expression
is only valid
+
2x + 1 x - 3
1
when x ¹ - and x ≠ 3.
2
24
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 4, continued
2. Find a common denominator.
The two denominators are 2x + 1 and x – 3. They
share no factors, so their product will be a good
common denominator.
(2x + 1)(x – 3) = 2x(x – 3) + 1(x – 3)
= 2x2 – 6x + x – 3
= 2x2 – 5x – 3
The common denominator is 2x2 – 5x – 3.
25
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 4, continued
3. Rewrite each term of the expression using
the new denominator.
3x
+
8
Original expression
2x + 1 x - 3
=
x-3
•
3x
+
2x + 1
•
8
x - 3 2x + 1 2x + 1 x - 3
Multiply
3x
by
x-3
2x + 1
x-3
8
2x + 1
and
by
.
x-3
2x + 1
26
3.1.1: Adding and Subtracting Rational Expressions
Guided Practice: Example 4, continued
=
3x(x - 3)
8(2x + 1)
+
(x - 3)(2x + 1) (2x + 1)(x - 3)
=
3x 2 - 9x
2x - 5x - 3
2
+
16x + 8
=
=
Multiply.
2x - 5x - 3
2
3x - 9x + (16x + 8)
2
2x 2 - 5x - 3
Simplify each term.
Rewrite the expression
by adding the
numerators over the
common denominator.
3x 2 + 7x + 8
2x - 5x - 3
2
3.1.1: Adding and Subtracting Rational Expressions
Combine like terms.
27
Guided Practice: Example 4, continued
4. Check to see if the result can be written
in a simpler form.
We already know the denominator has x – 3 and
2x + 1 as factors. The numerator, however, has no
integer factors. This expression cannot be written
in any other format that will simplify the result.
Thus
3x 2 + 7x + 8
, where x ¹ -
1
3.1.1: Adding and Subtracting Rational Expressions
✔
and x ≠ 3, is the
2
2x - 5x - 3
simplest way to rewrite the original
3x
8
expression,
+
.
2x + 1 x - 3
2
28
Guided Practice: Example 4, continued
29
3.1.1: Adding and Subtracting Rational Expressions