College AlgebraSixth EditionJames Stewart Lothar Redlin Saleem Watson

PrerequisitesP

Rational ExpressionsP.7

Fractional Expression

A quotient of two algebraic expressions

is called a fractional expression.

• Here are some examples:

2

2 2

1 4

x y

x y

3

2 25 6 1

x x x

x x x

Rational Expression

A rational expression is a fractional

expression in which both the numerator

and denominator are polynomials.

• Here are some examples:

3

2 2

2 2

1 4 5 6

x y x x

x y x x

Rational Expressions

In this section, we learn:

• How to perform algebraic operations on rational expressions.

The Domain of

an Algebraic Expression

In general, an algebraic expression may

not be defined for all values of the variable.

The domain of an algebraic expression is:

• The set of real numbers that the variable is permitted to have.

The Domain of an Algebraic Expression

The Domain of an Algebraic Expression

The table gives some basic expressions

and their domains.

E.g. 1—Finding the Domain of an Expression

Find the domains of these expressions.

2

2

a 2 3 1

b5 6

c5

x x

x

x x

x

x

E.g. 1—Finding the Domain

2x2 + 3x – 1

This polynomial is defined for every x.

• Thus, the domain is the set of real numbers.

Example (a)

E.g. 1—Finding the Domain

We first factor the denominator:

• Since the denominator is zero when x = 2 or x = 3.

• The expression is not defined for these numbers.

• The domain is: {x | x ≠ 2 and x ≠ 3}.

Example (b)

2 5 6 2 3

x x

x x x x

E.g. 1—Finding the Domain

• For the numerator to be defined, we must have x ≥ 0.

• Also, we cannot divide by zero, so x ≠ 5.• Thus the domain is {x | x ≥ 0 and x ≠ 5}.

Example (c)

5

x

x

Simplifying

Rational Expressions

Simplifying Rational Expressions

To simplify rational expressions, we

factor both numerator and denominator

and use following property of fractions:

• This allows us to cancel common factors from the numerator and denominator.

AC A

BC B

E.g. 2—Simplifying Rational Expressions by Cancellation

Simplify:

2

2

1

2

x

x x

2

2

)(Factor

(Cancel common factors)

1

1

21

2

2

1

1

x

x xxx

x

x

x

x

Caution

We can’t cancel the x2’s in

because x2 is not a factor.

2

2

1

2

x

x x

Multiplying and Dividing

Rational Expressions

Multiplying Rational Expressions

To multiply rational expressions, we use

the following property of fractions:

This says that: • To multiply two fractions, we multiply their

numerators and multiply their denominators.

A C AC

B D BD

E.g. 3—Multiplying Rational Expressions

Perform the indicated multiplication,

and simplify:

2

2

2 3 3 12

8 16 1

x x x

x x x

E.g. 3—Multiplying Rational Expressions

We first factor:

2

2

2

2

Factor

Property of fractions

Cancel common factors

2 3 3 12

8 16 11 3 3 4

14

3 3

3 3

41

41

4

x

x x x

x x xx x x

xx

x

x

x

x

x x

Dividing Rational Expressions

To divide rational expressions, we use

the following property of fractions:

This says that: • To divide a fraction by another fraction,

we invert the divisor and multiply.

A C A D

B D B C

E.g. 4—Dividing Rational Expressions

Perform the indicated division, and

simplify:2

2 2

4 3 4

4 5 6

x x x

x x x

E.g. 4—Dividing Rational Expressions

2

2 2

2

2 2Invert diviser and multiply

Factor

Cancel common factors

4 3 4

4 5 6

4 5 6

4 3 43

2 1

3

1

4

2 4

2

2

x x x

x x x

x x x

x x xx

x x

x

x

x

x

x

x

x

Adding and Subtracting

Rational Expressions

Adding and Subtracting Rational Expressions

To add or subtract rational expressions,

we first find a common denominator and

then use the following property of fractions:

A B A B

C C C

Adding and Subtracting Rational Expressions

Although any common denominator will work,

it is best to use the least common

denominator (LCD) as explained in Section

P.2.

• The LCD is found by factoring each denominator and taking the product of the distinct factors, using the highest power that appears in any of the factors.

Caution

Avoid making the following error:

A

B C

A A

B C

Caution

For instance, if we let A = 2, B = 1, and C = 1,

then we see the error:

?

?

?

2 2 2

1 1 1 12

2 22

1 4 Wrong!

E.g. 5—Adding and Subtracting Rational Expressions

Perform the indicated operations, and

simplify:

22

3(a)

1 2

1 2(b)

1 1

x

x x

x x

E.g. 5—Adding Rational Exp.

2

2

Fractions by LCD

Add fractions

Combine terms in numerator

3

1 23 2 1

1 2 1 2

3 6

1 2

2 6

1 2

x

x xx x x

x x x x

x x x

x x

x x

x x

Example (a)

Here LCD is simply the product (x – 1)(x + 2).

E.g. 5—Subtracting Rational Exp.

The LCD of x2 – 1 = (x – 1)(x + 1) and (x + 1)2

is (x – 1)(x + 1)2.

Example (b)

22

2

2

Factor

Combine fractions using LCD

1 2

1 1

1 2

1 1 1

1 2 1

1 1

x x

x x x

x x

x x

E.g. 5—Subtracting Rational Exp.

2

2

Distributive Property

Combine terms in numerator

1 2 2

1 1

3

1 1

x x

x x

x

x x

Example (b)

Compound Fractions

Compound Fraction

A compound fraction is:

• A fraction in which the numerator, the denominator, or both, are themselves fractional expressions.

E.g. 6—Simplifying a Compound Fraction

Simplify:

1

1

xy

yx

E.g. 6—Simplifying

One solution is as follows.

1. We combine the terms in the numerator into a single fraction.

2. We do the same in the denominator.

3. Then we invert and multiply.

Solution 1

E.g. 6—Simplifying

Thus,

1

1

x x yy y

y x yx x

x y x

y x y

x x y

y x y

Solution 1

E.g. 6—Simplifying

Another solution is as follows.

1. We find the LCD of all the fractions in the expression,

2. Then multiply the numerator and denominator by it.

Solution 2

E.g. 6—Simplifying

Here, the LCD of all the fractions is xy.

2

2

Multiply num.and denom.by

Simplify

Factor

1 1

1 1xy

x xy y

y yx x

x xy

xy y

x x y

y x y

xy

xy

Solution 2

Simplifying a Compound Fraction

The next two examples show situations

in calculus that require the ability to work

with fractional expressions.

E.g. 7—Simplifying a Compound Fraction

Simplify:

• We begin by combining the fractions in the numerator using a common denominator:

1 1a h a

h

E.g. 7—Simplifying a Compound Fraction

Combine fractions in numerator

Invert divisor and multiply

1 1

1

a h ah

a a h

a a h

ha a h

a a h h

E.g. 7—Simplifying a Compound Fraction

DistributiveProperty

Simplify

Cancel common factors

1

1

1

a a h

a a h h

a a h

a a h

h

h

E.g. 8—Simplifying a Compound Fraction

Simplify:

1 1

2 2 22 2

2

1 1

1

x x x

x

E.g. 8—Simplifying

Factor (1 + x2)–1/2 from the numerator.

1/21/2 1/2 2 2 22 2 2

2 2

1/22

2

3/22

1 11 1

1 1

1

11

1

x x xx x x

x x

x

x

x

Solution 1

E.g. 8—Simplifying

Since (1 + x2)–1/2 = 1/(1 + x2)1/2 is a fraction,

we can clear all fractions by multiplying

numerator and denominator by (1 + x2)1/2.

Solution 2

E.g. 8—Simplifying

1/2 1/22 2 2

2

1/2 1/2 1/22 2 2 2

1/22 2

2 2

3/2 3/22 2

1 1

1

1 1 1

1 1

1 1

1 1

x x x

x

x x x x

x x

x x

x x

Solution 2

Thus,

Rationalizing the Denominator

or the Numerator

Rationalizing the Denominator

If a fraction has a denominator of the form

we may rationalize the denominator by

multiplying numerator and denominator

by the conjugate radical .

A B C

A B C

Rationalizing the Denominator

This is effective because, by Product Formula

1 in Section P.5, the product of the

denominator and its conjugate radical does

not contain a radical:

2 2A B C A B C A B C

E.g. 9—Rationalizing the Denominator

Rationalize the denominator:

• We multiply both the numerator and the denominator by the conjugate radical of , which is .

1

1 2

1 2 1 2

Thus,

E.g. 9—Rationalizing the Denominator

22

Product Formula1

1 1

1 2 1 2

1 2

1 2

1 2

1 2

1 22 1

1 2 1

1 2

E.g. 10—Rationalizing the Numerator

Rationalize the numerator:

• We multiply numerator and denominator by the conjugate radical :

4 2h

h

4 2h

E.g. 10—Rationalizing the Numerator

Thus,

22

Product Formula1

4 2

4 2

4 2

4 2

4

4 2

2

h

h

h

h

h

h

h

h h

E.g. 10—Rationalizing the Numerator

Cancel common factors

4 4

4 2

4 2

1

4 2

h

h h

h

h h

h

Avoiding Common Errors

Avoiding Common Errors

Don’t make the mistake of applying

properties of multiplication to the

operation of addition.

• Many of the common errors in algebra involve doing just that.

Avoiding Common Errors

The following table states several

multiplication properties and illustrates the

error in applying them to addition.

Avoiding Common Errors

To verify that the equations in the right-hand

column are wrong, simply substitute numbers

for a and b and calculate each side.

Avoiding Common Errors

For example, if we take a = 2 and b = 2

in the fourth error, we get different values for

the left- and right-hand sides:

Avoiding Common Errors

The left-hand side is:

The right-hand side is:

• Since 1 ≠ ¼, the stated equation is wrong.

1 1 1 1

12 2a b

1 1 1

2 2 4a b

Avoiding Common Errors

You should similarly convince yourself

of the error in each of the other equations.• See Exercise 119.