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Section A.3
Rational Expressions• Goals
– Factor and simplify rational expressions– Simplify complex fractions– Rationalize an expression involving radicals– Simplify an algebraic expression with negative
exponents
Rational Expressions
• A rational expression is a quotient p/q of two polynomials p and q .
• When simplifying a rational expression, begin by factoring out the greatest common factor in the numerator and denominator.
• Next, factor the numerator and denominator completely. Then cancel any common factors.
Domain and Examples• The domain of a rational expression p/q
consists of all real numbers except those that make the denominator zero…– …since division by zero is never allowed!
• Examples:
Example• Simplify the expression:
225 1005 10tt
22
Solution:
25 4 25 2 ( 2)25 100
5 10 5( 2) 5( 2)
5 25
t t tt
t t t
2 ( 2)t t
5 ( 2)t 5( 2)t
(Solution will appear when you click)
Another Example• Simplify the expression:
34 12
xx
Solution:
3 3
4 12 4( 3)
Note that 1( ) .
Thus, 3 1(3 ).
3 3So we have
4( 3)
x x
x x
a b b a
x x
x x
x
4 1(3 )x 1
.4
Simplifying a Complex Fraction
52
Example: Simplify .1
43
x
x
To simplify a complex fraction, multiply the entire numerator and denominator by the least common denominator of the inner fractions.
The inner fractions are and .
Their LCD is
Thus, we should multiply the entire numerator and denominator by
5
x
1
3x
3 .x
3 .x
Solution, continued
5 52 3 3 2 3
11 3 4 34 333
x x xx x
x xxxx
5
x3 x 2 3
1
3
x
x
3x
15 6
1 124 3
x
xx
Expressions with Negative Exponents
• Recall that
• If the expression contains negative exponents, one way to simplify is to rewrite the expression as a complex fraction.
• Recall the following properties of exponents:
1nnxx
anda
a b a b a bb
xx x x x
x
Example: Simplify
2 5
3 2
x yx y
22 5 5
3 2
3 2
1
1 1
xx y yx y
x y
The least common denominator of the inner fractions is
3 5.x y
3 52 3 5 3 5 2 3 5
5 5
3 5 3 53 5 3 5
3 2 3 2
1
1 1
x yx x y x y x y
y yx y x yx y x y
x y x y
3 55 5
x yx y
5y3x 5
3
y
x
5 5 3
5 3 33 5 2
x y xy x y
x y
First, rewrite the expression as a complex fraction.
Multiply numerator and denominator by the LCD.
Rationalizing the Denominator
• Given a fraction whose denominator is of the form or we sometimes want to rewrite the fraction with no square roots in the denominator.
• This is called rationalizing the denominator of the given fractional expression.
• It often allows the fraction to be simplified.
a b a b
Rationalizing (cont’d)
• To rationalize the denominator, multiply both numerator and denominator of the fraction by the conjugate of the denominator.
• The conjugate of is
the conjugate of is
a b ;a b .a ba b
Example• Rationalize the denominator of
• Note that we can also rationalize numerators in the same way!
1
x y
Another Example• Rationalize the denominator of
2 2b c
b c
2 2
2 2
Multiply numerator and denominator
by the conjugate of the denominator.
Factor the difference of squares.
Multiply out the denominator.
b c b c
b c b c
b c b c
b c b c
b c
( )b c b c
b c
( )b c b c
ExampleWrite the expression as a single quotient in which only positive exponents and/or radicals appear:
Assume that
1/2 1/25 7 5
5
x x x
x
5.x
First, notice that the numerator has a common factor of x + 5.
Always take the lower exponent when taking out a common factor.In this case, the exponents are 1/2 and -1/2, so the lower exponent is -1/2.
1/2 1/25 7 5
5
x x x
x
To find the exponent that is left when you take out a common factor to a certain power, subtract that power from each exponent of the common factor.Remember, we are factoring out
continued on next slide. . .
1 1/2 1/2/2 ( 1/2) ( 1/2)5 5 7 5
5
x x x x
x
1/25 .x
1/2 ( 1/2) ( 1/21/2 1/2
1/2 1 0
)5 5 7 5
5
5 5 7 5
5
x x x x
x
x x x x
x
1/2
5 7
5 5
x x
x x
3/2
5 6( 5)
xx
ExampleWrite the expression as a single quotient in which only positive exponents and/or radicals appear:
Assume that
1/2 1/242
3x x
0.x
Again, factor out the common factor with the lower exponent:
Now combine the expressions to form a single quotient.
1/2 (1/2 1/2 1/2 11 //2) ( 1/2 2)4 42 2
3 3x x x x x
1/2x
1/2 0 1 1/24 42 2
3 3x x x x x
1 42
3x
x
1 4 1 2 3 4
23 1 3 3x x
x x
1 6 4 6 43 3
x x
x x