Download 6.1 The Fundamental Property of Rational Expressions

6.1 The Fundamental
Property of Rational
Expressions
The Fundamental Property of Rational Expressions
The quotient of two integers (with the denominator not 0), such as
2
3
or  3 , is called a rational number. In the same way, the quotient of two
4
polynomials with the denominator not equal to 0 is called a rational
expression.
Rational Expression
P
,
A rational expression is an expression of the form
where
P and Q are
Q
polynomials, with Q ≠ 0.
6 x
,
3
x 8
9x
,
y3
2m3
8
Examples of rational
expressions
Slide 6.1-3
Objective 1
Find the numerical value of a rational
expression.
Slide 6.1-4
CLASSROOM
EXAMPLE 1
Evaluating Rational Expressions
Find the value of the rational expression, when x = 3.
x
2x 1
Solution:
3


2  3  1
3

6 1
3

7
Slide 6.1-5
Objective 2
Find the values of the variable for
which a rational expression is
undefined.
Slide 6.1-6
Find the values of the variable for which a rational
expression is undefined.
P
In the definition of a rational expression Q ,Q cannot equal 0.
The denominator of a rational expression cannot equal 0 because division
by 0 is undefined.
For instance, in the rational expression
3x  6
2x  4
Denominator cannot equal 0
the variable x can take on any real number value except 2. If x is 2, then the
denominator becomes 2(2) − 4 = 0, making the expression undefined. Thus, x
cannot equal 2. We indicate this restriction by writing x ≠ 2.
Since we are solving to find values that make the expression undefined,
we write the answer as “variable ≠ value”, not “variable = value or { } .
Slide 6.1-7
Find the values of the variable for which a rational expression
is undefined. (cont’d)
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. The variable cannot equal these
values.
The numerator of a rational expression may be any real number. If the
numerator equals 0 and the denominator does not equal 0, then the rational
expression equals 0.
Slide 6.1-8
CLASSROOM
EXAMPLE 2
Finding Values That Make Rational Expressions Undefined
Find any values of the variable for which each rational expression is
undefined.
Solution:
x2
x5
x 5  0
x 55  05
x5
3r
3r

r 2  6r  8
 r  2 r  4
5z 1
z2  5
r20
r 22  02
r40
r 44  04
r  2
r  4
never undefined
Slide 6.1-9
Objective 3
Write rational expressions in lowest
terms.
Slide 6.1-10
Write rational expressions in lowest terms.
A fraction such as 2 is said to be in lowest terms.
3
Lowest Terms
P
A rational expression Q (Q ≠ 0) is in lowest terms if the greatest common
factor of its numerator and denominator is 1.
Fundamental Property of Rational Expressions
If P(Q ≠ 0) is a rational expression and if K represents any polynomial,
Q
where K ≠ 0, then
PK P
 .
QK Q
This property is based on the identity property of multiplication, since
PK P K P
P
   1  .
QK Q K Q
Q
Slide 6.1-11
CLASSROOM
EXAMPLE 3
Writing in Lowest Terms
Write each rational expression in lowest terms.
Solution:
15
45
35

335
6 p3
2 p2
2 3 p  p  p

2 p  p
1

3
 3p
Slide 6.1-12
Write rational expressions in lowest terms. (cont’d)
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.
Quotient of Opposites
If the numerator and the denominator of a rational expression are opposites,
as in
x y
y  x then the rational expression is equal to −1.
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.
Numerator cannot
6 x
6 x  9 3  2 x  3 3
be factored.


4x  6
2  2 x  3
2
4x
Slide 6.1-13
CLASSROOM
EXAMPLE 4
Writing in Lowest Terms
Write in lowest terms.
Solution:
2  2 y  1
4y  2
6y  3

a 2  b2
a 2  2ab  b 2
a  b  a  b 


 a  b  a  b 
3  2 y  1
2

3
a  b


a  b
Slide 6.1-14
CLASSROOM
EXAMPLE 5
Write
Writing in Lowest Terms (Factors Are Opposites)
z 2  5 in lowest terms.
5  z2
Solution:
1 z  5 
2

1 z  5 
2
 1
Slide 6.1-15
CLASSROOM
EXAMPLE 6
Writing in Lowest Terms (Factors Are Opposites)
Write each rational expression in lowest terms.
Solution:
5 y
y 5
1 y  5

y 5
25 x 2  16
12  15 x
5 x  4  5 x  4 
5x  4




3  5 x  4 
3
9k
9k
already in lowest terms
 1
5x  4
or 
3
Slide 6.1-16
Objective 4
Recognize equivalent forms of rational
expressions.
Slide 6.1-17
Recognize equivalent forms of rational expressions.
When working with rational expressions, it is important to be able to
recognize equivalent forms of an expressions. For example, the common
5
5

5

fraction 6 can also be written
and 6 . Consider the rational
6
expression
2x  3

.
2
The − sign representing the factor −1 is in front of the expression, even
with fraction bar. The factor −1 may instead be placed in the numerator or
in the denominator. Some other equivalent forms of this rational
expression are
  2 x  3
2
and
2x  3
2
Slide 6.1-18
Recognize equivalent forms of rational expressions. (cont’d)
By the distributive property,
  2 x  3
2
can also be written
2 x  3
.
2
  2 x  3
2 x  3
is not an equivalent form of
. The sign preceding 3 in
2
2
2 x  3
the numerator of
should be − rather than +. Be careful to apply
2
the distributive property correctly.
Slide 6.1-19
CLASSROOM
EXAMPLE 7
Writing Equivalent Forms of a Rational Expression
Write four equivalent forms of the rational expression.
2x  7

x3
Solution:
  2x  7
,
x3
2 x  7
,
x3
2x  7
,
  x  3
2x  7
x  3
Slide 6.1-20