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