Download Rational Expressions and Replacements

Math 52 (Summer ’09)
6.1 "Multiplying and Simplifying Rational Expressions"
Objectives:
*
Find all numbers for which a rational expression is not de…ned.
*
*
A
Multiply a rational expression by 1, using an expression such as :
A
Multiply and Simplify rational expressions.
Rational Expressions and Replacements
The following are called rational expressions or fraction expressions. They are quotients, or ratios, of polynomials:
:
A rational expression is also a division. For example,
. Because rational
expressions indicate division, we must be careful to avoid denominators of zero. When a variable is replaced with a number
that produces a denominator equal to zero, the rational expression is not de…ned.
Example 1: (Finding values for which the rational expressions are not de…ned)
Find all numbers for which the rational expression is not de…ned.
7 3x + x2
x2 9
b)
a)
2x 8
49 x2
Multiplying By 1
Multiplying Rational Expressions:
To multiply rational expressions, multiply numerators and denominators:
Expressions that have the same value for all allowable replacements are called equivalent expressions. We can multiply
by 1 to obtain an equivalent expression.
Example 2: (Multiplying by 1)
Multiply, but do not simplify.
2x x 1
a)
2x x + 4
b)
Page: 1
2a 3
5a + 2
a
a
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
6.1
Simplifying Rational Expressions
Example 3: (Simplifying rational expressions)
Simplify:
4x 12
a)
4x
c)
8y 2
4y 2
b)
32
16
CAUTION!!
t2 9
+ 5t + 6
t2
4x + 32
+ 9x + 8
d)
x2
The di¢ culty with canceling is that it is often applied incorrectly, as in the following situations:
Opposites in Rational Expressions:
Expressions of the form
binomials is multiplied by
and
are opposites of each other. When either of these
1, the result is the other binomial:
and
:
Example 4: (Opposites in rational expressions)
Simplify:
x 8
a)
8 x
b)
Page: 2
6t
2
12
t
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
6.1
Multiply and Simplifying
We try to simplify after we multiply. That is why we leave the numerator and the denominator in factored form.
Example 5: (Multiplying and simplifying)
Multiply and simplify:
5a3 2
a)
4 5a
c)
x4
x4
1 x2 + 9
81 x2 + 1
b)
d)
Page: 3
x2
x2
4x + 4 x + 3
9
x 2
5w2 180 20w + 20
10w2 10 2w 12
Notes by Bibiana Lopez