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7.1 – Slide 1
Chapter 7
Rational Expressions and
Applications
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7.1 – Slide 2
1
7.1
The Fundamental Property of
Rational Expressions
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7.1 – Slide 3
7.1 The Fundamental Property of Rational
Expressions
Objectives
1.
2.
3.
4.
Find the values of the variable for which a rational
expression is undefined.
Find the numerical value of a rational expression.
Write rational expressions in lowest terms.
Recognize equivalent forms of rational
expressions.
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7.1 – Slide 4
2
7.1 The Fundamental Property of Rational
Expressions
Rational Expression
A rational expression is an expression of the form
P
,
Q
where P and Q are polynomials, with Q ≠ 0.
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7.1 – Slide 5
7.1 The Fundamental Property of Rational
Expressions
Determining When a Rational Expression is Undefined
Determining When a Rational Expression is Undefined
Step 1 Set the denominator of the rational expression
equal to 0.
Step 2 Solve this equation.
Step 3 The solutions of the equation are the values that
make the rational expression undefined.
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7.1 – Slide 6
3
7.1 The Fundamental Property of Rational
Expressions
Determining When a Rational Expression is Undefined
Example 1
Find any values of the variable for which each rational
expression is undefined.
x+3
(a) 2
x − 10 x + 9
x2 – 10x + 9 = 0
Set the denominator equal to 0.
Factor.
(x – 1)(x – 9) = 0
x–1=0
x=1
or
x–9=0
x=9
Zero-factor property
Solve.
The original expression is undefined for x = 1 or x = 9.
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7.1 – Slide 7
7.1 The Fundamental Property of Rational
Expressions
Determining When Rational Expressions are Undefined
Example 1 (concluded)
Find any values of the variable for which each rational
expression is undefined.
(b)
y3
y 2 + 25
Set the denominator equal to 0 and solve.
y2 + 25= 0
The denominator will not equal 0 for any value of y because y2
is always greater than or equal to 0, and adding 1 makes the
sum greater than 0.
Thus, there are no values for which this rational expression
is undefined.
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7.1 – Slide 8
4
7.1 The Fundamental Property of Rational
Expressions
Finding the Numerical Value of a Rational Expression
Example 2
2z + 7
Find the numerical value of 2
for each value of z.
z − 36
(a) z = –1
(b) z = 6
2z + 7
2(−1) + 7
2z + 7
2(6) + 7
=
=
2
2
2
z − 36 (−1) − 36
z − 36 (6)2 − 36
−2 + 7
12 + 7
=
=
1 − 36
36 − 36
5
19
=
=
−35
0
1
The
expression
is undefined
=−
when z = 6.
7
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7.1 – Slide 9
7.1 The Fundamental Property of Rational
Expressions
Writing Rational Expressions in Lowest Terms
Lowest Terms
P
A rational expression
(Q ≠ 0) is in lowest terms if the
Q
greatest common factor of its numerator and denominator
is 1.
Fundamental Property of Rational Expressions
P
(Q ≠ 0) is a rational expression and if K represents
Q
any polynomial, where K ≠ 0, then
PK P
= .
QK Q
If
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7.1 – Slide 10
5
7.1 The Fundamental Property of Rational
Expressions
Writing Rational Expressions in Lowest Terms
Writing a Rational Expression in Lowest Terms
Step 1 Factor the numerator and denominator completely.
Step 2 Use the fundamental property to divide out any
common factors.
7.1 – Slide 11
Copyright © 2010 Pearson Education, Inc. All rights reserved.
7.1 The Fundamental Property of Rational
Expressions
Writing Rational Expressions in Lowest Terms
Example 3
Write each rational expression in lowest terms.
(a)
8 x + 24
8( x + 3)
=
10 x + 30 10( x + 3)
8
10
4
=
5
=
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(b)
3y2 − 6 y
3 y ( y − 2)
=
2
y − y − 2 ( y + 1)( y − 2)
=
3y
y +1
7.1 – Slide 12
6
7.1 The Fundamental Property of Rational
Expressions
Writing Rational Expressions in Lowest Terms
CAUTION
Rational expressions cannot be written in lowest terms
until after the numerator and denominator have been
factored. Only common factors can be divided out, not
common terms. For example,
6 x + 9 3( 2 x + 3) 3
=
=
4 x + 6 2(2 x + 3) 2
Divide out the
common factor.
6+ x
4x
Numerator cannot
be factored
This expression is already in
lowest terms.
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7.1 – Slide 13
7.1 The Fundamental Property of Rational
Expressions
Writing Rational Expressions in Lowest Terms
Example 4
Write the rational expression in lowest terms.
x− y
At first, it does not appear that the numerator
y−x
and denominator share any common factor.
However, use the fact that the denominator, y – x, can be
rewritten as
y – x = –1(–y +x) = –1(x – y)
Thus,
x− y
x− y
=
y − x −1( x − y )
=
1
= −1
−1
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7.1 – Slide 14
7
7.1 The Fundamental Property of Rational
Expressions
Writing Rational Expressions in Lowest Terms
If the numerator and denominator of a rational expression
x−y
, then the rational
are opposites, such as in
y −x
expression is equal to –1.
CAUTION
Although x and y appear in both the numerator and
denominator in Example 3(c), we cannot use the
fundamental property right away because they are terms,
not factors. Terms are added, while factors are
multiplied.
7.1 – Slide 15
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7.1 The Fundamental Property of Rational
Expressions
Recognizing Equivalent Forms of Rational Expressions
Example 5
Write three equivalent forms of the rational expression.
−
x2
2x − 5
−x2
1.
2x − 5
Consider different placements for the – sign.
x2
2.
−(2 x − 5)
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x2
x2
3.
=
−2 x + 5 5 − 2 x
7.1 – Slide 16
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