the fundamental property of rational expressions find the numerical value of a rational expression....

The Fundamental Property of Rational Expressions

Find the numerical value of a rational expression.

Find the values of the variable for which a rational expression is undefined.

Write rational expressions in lowest terms.

Recognize equivalent forms of rational expressions.

7.1

2

3

4

1

Rational ExpressionA rational expression is an expression of the form where P and Q are polynomials, with Q ≠ 0.

,P

Q

9,

3

x

y

32

8

m3

6,

8

x

x

Examples of rational expressions

The Fundamental Property of Rational Expressions

The quotient of two integers (with the denominator not 0), such as

or is called a rational number. In the same way, the quotient

of two polynomials with the denominator not equal to 0 is called a

rational expression.

2

3

3,

4

Find the value of the rational expression, when x = 3.

Solution:

2 1

x

x

1

3

2 3

3

6 1

3

7

EXAMPLE 1 Evaluating Rational Expressions

Find the values of the variable for which a rational expression is undefined.

In the definition of a rational expression Q cannot equal 0.

The denominator of a rational expression cannot equal 0 because

division by 0 is undefined.

,P

Q

Since we are solving to find values that make the expression undefined, we write the answer as “variable ≠ value”, not “variable = value or { } .

For instance, in the rational expression

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.

3 6

2 4

x

x

Denominator cannot equal 0

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 7.1-8

Find the values of the variable for which a rational expression is undefined. (cont’d)

Find any values of the variable for which each rational expression is undefined.

Solution:

2

5

x

x

2

3

6 8

r

r r

2

5 1

5

z

z

5 0x

2 0r

5 55 0x 5x

3

2 4

r

r r

2 22 0r

2r

4 0r 4 44 0r

4r

never undefined

EXAMPLE 2 Finding Values That Make Rational Expressions Undefined

A fraction such as is said to be in lowest terms.

Write rational expressions in lowest terms.

2

3

Fundamental Property of Rational ExpressionsIf (Q ≠ 0) is a rational expression and if K represents any

polynomial, where K ≠ 0, then .

PK P

QK Q

P

Q

Lowest TermsA rational expression (Q ≠ 0) is in lowest terms if the greatest

common factor of its numerator and denominator is 1.

P

Q

1 .K K

K K

P P P P

Q Q Q Q

This property is based on the identity property of multiplication, since

Solution:

3

2

6

2

p

p

Write each rational expression in lowest terms.

15

45

32

2

p pp

p p

3 5

3 53

3p

1

3

EXAMPLE 3 Writing in Lowest Terms

Quotient of Opposites

If the numerator and the denominator of a rational expression are

opposites, as in then the rational expression is equal to −1.x y

y x

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.

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.

2 3

2

36 9 3

4 6 2 23

x

x

x

x

6

4

x

x

Numerator cannot be factored.

Write rational expressions in lowest terms. (cont’d)

Solution:

2

3

2 12

3 2 1

y

y

4 2

6 3

y

y

a ba b

a a bb

a b

a b

Write in lowest terms.

2 2

2 22

a b

a ab b

EXAMPLE 4 Writing in Lowest Terms

Write in lowest terms.2

2

5

5

z

z

Solution:

2

2

5

5

1

1

z

z

1

EXAMPLE 5 Writing in Lowest Terms (Factors Are Opposites)

Write each rational expression in lowest terms.

5

5

y

y

225 16

12 15

x

x

9

9

k

k

5

5

1 y

y

5 4

3

5 4

5 4

xx

x

already in lowest terms

5 4

3

x

1

5 4

3

x or

EXAMPLE 6 Writing in Lowest Terms (Factors Are Opposites)

Solution:

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 fraction can also be written and Consider the

rational expression2 3

.2

x

5.

65

6

5

6

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

and 2 3

2

x 2 3

2

x

is not an equivalent form of . The sign preceding 3 in

the numerator of should be − rather than +. Be careful to apply

the distributive property correctly.

2 3

2

x 2 3

2

x

2 3

2

x

By the distributive property,

can also be written 2 3

2

x 2 3.

2

x

Recognize equivalent forms of rational expressions. (cont’d)

Write four equivalent forms of the rational expression.

2 7

3

x

x

2 7,

3

x

x

Solution:

2 7

,3

x

x

2 7,

3

x

x

2 7

3

x

x

EXAMPLE 7 Writing Equivalent Forms of a Rational Expression

Multiplying and Dividing Rational Expressions

Multiply rational expressions.

Divide rational expressions.

7.2

2

1

Multiply rational expressions.

Multiplying Rational Expressions

The product of the rational expressions and is

That is, to multiply rational expressions, multiply the numerators and multiply the denominators.

P

QR

SP R PR

Q S QS

The product of two fractions is found by multiplying the numerators and multiplying the denominators. Rational expressions are multiplied in the same way.

Multiply. Write each answer in lowest terms.

Solution:

2 5

7 10

2

2

8 9

3

p q

pq

2 5

7 10

2

2

8 9

3

p q

p q

7

2 5

2 5

1

7

38

3

3p qp

p q q

24p

q

It is also possible to divide out common factors in the numerator and denominator before multiplying the rational expressions.

EXAMPLE 1 Multiplying Rational Expressions

Multiply. Write the answer in lowest terms.

Solution:

3

2

p q q

p p q

3

2

p q q

p q p

3

2

q

q p

p q

p

3

2

q

p


Multiply. Write the answer in lowest terms.

2

2

7 10 6 6

3 6 2 15

x x x

x x x

Solution:

2

2

7 10 6 6

3 6 2 15

x x x

x x x

2 13 2

3 5 3

5

2

x x x

xx x

2 1

3

x

x

2 6 6

3 3

5

56

x x

x

x

x x


Divide rational expressions.

Dividing Rational Expressions

If and are any two rational expressions with then

That is, to divide one rational expression by another rational expression, multiply the first rational expression by the reciprocal of the second rational expression.

.P R P S PS

Q S Q R QR

0,R

S

R

S

P

Q

Solution:

2 9.y x

4

3 44

5

3 16

4 5 12

5

3

2

3

3

p p

pp p

3 5

4 16

2

3

3 4

3 4

9

6

p p

p p

Divide. Write each answer in lowest terms.

2 39 6

3 4 3 4

p p

p p

3

2p

EXAMPLE 4 Dividing Rational Expressions

Divide. Write the answer in lowest terms.

Solution:

2 25 10

2 8

a b ab

2

2

5 8

2 10

a b

ab 5 2 2

2 2 5

2a b

a

a

bb

2a

b


Divide. Write the answer in lowest terms.

2

2

4 3 3

2 1 4 1

x x x x

x x

2

2

4 4

1

3

2 3

1xx x

x x x

2 1

1

4 2

2

1

1

x x

xx

x

x

4 2 1x

x

Solution:


2

2 21 2 1

ab a a b

a a a

Divide. Write in the answer in lowest terms.

Solution:

1 1

1

1

1

a aa b a

a ba a

2 2

2

2 1

1

ab a a a

a a b

1

1

a a

a

Remember to write −1 when dividing out factors that are opposite of each other. It may be written in the numerator or denominator, but not both.

EXAMPLE 7 Dividing Rational Expressions (Factors Are Opposites)

Multiplying or Dividing Rational Expressions.

Multiplying or Dividing Rational Expressions

Step 1: Note the operation. If the operation is division, use the definition of division to rewrite it as multiplication.

Step 3: Multiply numerators and denominators.

Step 2: Factor all numerators and denominators completely.

Step 4: Write in lowest terms using the fundamental property.

Least Common Denominators

Find the least common denominator for a group of fractions.

Write equivalent rational expressions.

7.3

2

1

Find the least common denominator for a group of fractions.

Adding or subtracting rational expressions often requires a least

common denominator (LCD), the simplest expression that is

divisible by all of the denominators in all of the expressions. For

example, the least common denominator for the fractions and

is 36, because 36 is the smallest positive number divisible by both 9

and 12.

2

9

5

12

We can often find least common denominators by inspection. For

example, the LCD for and is 6m. In other cases, we find the

LCD by a procedure similar to that used for finding the greatest

common factor.

1

6

2

3m

Finding the Least Common Denominator (LCD)

Step 1: Factor each denominator into prime factors.

Step 2: List each different denominator factor the greatest number of times it appears in any of the denominators.

Step 3: Multiply the denominator factors from Step 2 to get the LCD.

When each denominator is factored into prime factors, every prime factor must be a factor of the least common denominator.

Find the least common denominator for a group of fractions. (cont’d)

Find the LCD for each pair of fractions.

Solution:

7 1,

10 25

4 6

4 11,

8 12m m

10 2 5

4 48 2 2 2m m

25 5 5

6 612 2 2 3m m

432 m 622 3 m

52 25

2LCD 5 2

3 63D 2LC m

50

624m

EXAMPLE 1 Finding the LCD

Find the LCD for

Solution:

3 5

4 5 and .

16 9m n m

3 316 2 2 2 2m n m n 5 59 3 3m m

342 nm 2 53 m

4 2 52 3LCD m n 5144m n

When finding the LCD, use each factor the greatest number of times it appears in any single denominator, not the total number of times it appears.

EXAMPLE 2 Finding the LCD

Solution:

4 1,

1 1x x

Find the LCD for the fractions in each list.

2 2

6 3 1,

4 16

x

x x x

4x x

4 4x x

LCD 4 4x x x

4x x

44 xx

Either x − 1 or 1 − x, since they are opposite expressions.

EXAMPLE 3 Finding LCDs

Write equivalent rational expressions.

Writing A Rational Expression with a Specified Denominator

Step 1: Factor both denominators.

Step 2: Decide what factor (s) the denominator must be multiplied by in order to equal the specified denominator.

Step 3: Multiply the rational expression by the factor divided by itself. (That is, multiply by 1.)

Rewrite each rational expression with the indicated denominator.

Solution:

3

4

93

4 9

3

4 9

?

4

242

30

k

k

3 ?

4 36

7 ?

5 65

k

k

7 7

5 5

6

6

k k k

k 7 ?

5 30

k

k

27

36

EXAMPLE 4 Writing Equivalent Rational Expressions

Rewrite each rational expression with the indicated denominator.

Solution:

9 ?

2 5 6 15a a

2

5 1 ?

2 2 1

k

k k k k k

9 ?

32 5 2 5a a

9 9

2 5 2

3

35a a

27

6 15a

1

5 1 ?

2 2

k

k k kk k

5 1 5 1 1

2 12

k k

k k k k

k

k

5 1 1

2 1

k k

k k k

EXAMPLE 5 Writing Equivalent Rational Expressions

Adding and Subtracting Rational Expressions

Add rational expressions having the same denominator.

Add rational expressions having different denominators.

Subtract rational expressions.

7.4

2

3

1

Add rational expressions having the same denominator.

We find the sum of two rational expressions with the same procedure used for adding two fractions having the same denominator.

Adding Rational Expressions (Same Denominator)

If and (Q ≠ 0) are rational expressions, then

That is, to add rational expressions with the same denominator, add the numerators and keep the same denominator.

.P R P R

Q Q Q

P

Q

R

Q

Add. Write each answer in lowest terms.

Solution:

7 3

15 15

2 2x y

x y x y

7 3

15

10

15

5

2

3

5

2

3

2 2x y

x y

2 x y

x y

2

EXAMPLE 1 Adding Rational Expressions (Same Denominator)

We use the following steps to add fractions having different denominators.

Add rational expressions having different denominators.

Adding Rational Expressions (Different Denominators)

Step 1: Find the least common denominator (LCD).

Step 2: Rewrite each rational expression as an equivalent rational expression with the LCD as the denominator.

Step 3: Add the numerators to get the numerator of the sum. The LCD is the denominator of the sum.

Step 4: Write in lowest terms using the fundamental property.

Add. Write each answer in lowest terms.

Solution:

1 1

10 15

2

3 7

m

n n

10 52

LCD 3 7 21n n

15 53 LCD 2 3 5 30

3 2

30 2

1 1

1 15

3 2

30 30

5

30

1

6

7 3

7

2

33 7

m

n n

7 6

21 21

m

n n

7 6

21

m

n

3 2

30

EXAMPLE 2 Adding Rational Expressions (Different Denominators)

Solution:

2

2 4

1 1

p

p p

Add. Write the answer in lowest terms.

2 4

1 1 1

p

p p p

2 2

1 1

p

p p

2 4

1

1

11 1

p p

ppp p

2 2 4

1 1

p p

p p

12

11p

p

p

2

1p

EXAMPLE 3 Adding Rational Expressions

Solution:

2 2

2 3

5 4 1

k

k k k

2 3

4 1 1 1

k

k k k k


1 4

1

2 3

4 1 41 1

k

k k k

k

kk

k

k

2 1 3 4

4 1 1 4 1 1

k k k

k k k k k k

22 2 3 12

4 1 1

k k k

k k k

22 5 12

4 1 1

k k

k k k

2 3 4

4 1 1

k k

k k k

EXAMPLE 4 Adding Rational Expressions


Solution:

2 3 3 2

m n

m n n m

2 3 3 2

1

1

m n

m n n m

2 3 3 2

m n

m n n m

or 2 3 3 2

m n n m

m n n m

2 3 2 3

m n

m n m n

EXAMPLE 5 Adding Rational Expressions (Denominators Are Opposites)

We subtract rational expressions having different denominators using a procedure similar to the one used to add rational expressions having different denominators.

Subtract rational expressions.

Subtracting Rational Expressions (Same Denominator)

If and (Q ≠ 0) are rational expressions, then

That is, to subtract rational expressions with the same denominator, subtract the numerators and keep the same denominator.

P R P R

Q Q Q

R

Q

R

Q

5 5

1 1

t t

t t

5 5

1

t t

t

5 5

1

t t

t

4 5

1

t

t

Solution:

Sign errors often occur in subtraction problems. The numerator of the fraction being subtracted must be treated as a single quantity. Be sure to use parentheses after the subtraction sign.

EXAMPLE 6 Subtracting Rational Expressions (Same Denominator)

Subtract. Write the answer in lowest terms.

6 1

2 3a a


Solution:

36 1

2 3

2

3 2

a a

a aa a

6 18 2

2 3 3 2

a a

a a a a

6 18 2

2 3

a a

a a

6 18 2

2 3

a a

a a

5 20

2 3

a

a a

5 4

2 3

a

a a

EXAMPLE 7 Subtracting Rational Expressions (Different Denominators)

4 3 1

1 1

x x

x x


Solution:

4 3 1

1 1

x x

x x

14 3 1

1 1 1

x x

x x

4 3 1

1

x x

x

4 3 1

1

x x

x

1

1

x

x

1

EXAMPLE 8 Subtracting Rational Expressions (Denominators Are Opposites)

2 2

3 4

5 10 25

r

r r r r

Solution:

3 4

5 5 5

r

r r r r

3 4

55 5

5

5

rr

r r

r

r rr r

23 15 4

5 5 5 5

r r r

r r r r r r

23 19

5 5

r r

r r r

3 19

5 5

r

r

r

r r

2

3 19

5

r

r

EXAMPLE 9 Subtracting Rational Expressions


Complex Fractions

Simplify a complex fraction by writing it as a division problem (Method 1).

Simplify a complex fraction by multiplying numerator and denominator by the least common denominator (Method 2).

7.5

2

1

Complex Fractions.

The quotient of two mixed numbers in arithmetic, such as can be written as a fraction.

In algebra, some rational expressions also have fractions in the numerator, or denominator, or both.

Complex Fraction

A quotient with one or more fractions in the numerator, or denominator, or both is called a complex fraction.

1 12 3

2 4

1 12 21 1 2 22 3

1 12 4 3 34 4

12

21

34

The parts of a complex fraction are named as follows.

Numerator of complex fractionMain fraction barDenominator of complex fraction

Since the main fraction bar represents division in a complex fraction, one method of simplifying a complex fraction involves division.

Simplify a complex fraction by writing it as a division problem (Method 1).

Method 1 for Simplifying a Complex Fraction

Step 1: Write both the numerator and denominator as single fractions.

Step 2: Change the complex fraction to a division problem.

Step 3: Perform the indicated division.

Simplify each complex fraction.

Solution:

2 15 41 12 3

12

6 34

m

mm

4 54 5

2 15 41 12

3 23 23

12

3 2

22

1

4

m

m

m

8 520 20

3 26 6

2 12 2

3 2 1

4

m

m

m

132056

13 5

20 6

13 6

20 5 78

100

39

0

2

52

39

50

2 12

3 2 1

4

m

m

m

3 2 12 1

2 4

mm

m

2 1 2

2 2 1

2

3

mm

m

2

3

m

EXAMPLE 1 Simplifying Complex Fractions (Method 1)

Simplify the complex fraction.

Solution:

2 3

4

2

m npm np

2 3 4

2

m n m n

p p

2 3 2

4

m n p

p m n

3 22

4

m

p

n p

nm

2

2

n p

m

m m n pn n p

p m m mm n

EXAMPLE 2 Simplifying a Complex Fraction (Method 1)

Solution:

1 21 2

2 12 3

a b

b a


2 1

2 11 2

1 2

2 13

3 2

3 22

b a

b a b

b

a

a b

a a b

2 2 2

1 2

2 6 2

2 3

b a

a b

a b

b a

2 2 8

1 2 2 3

a b a b

a b b a

32

1 2 8

2

2

b aa b

a bba

2 3

1 2 8

a b a

a a b


Since any expression can be multiplied by a form of 1 to get an equivalent expression, we can multiply both the numerator and denominator of a complex fraction by the same nonzero expression to get an equivalent rational expression. If we choose the expression to be the LCD of all the fractions within the complex fraction, the complex fraction will be simplified.

Simplify a complex fraction by multiplying numerator and denominator by the least common denominator (Method 2).

Method 2 for Simplifying a Complex Fraction

Step 1: Find the LCD of all fractions within the complex fraction.

Step 2: Multiply both the numerator and denominator of the complex fraction by this LCD using the distributive property as necessary. Write in lowest terms.

Solution:

2 13 44 19 2


2 13 4

3

61

6

42

39

24 9

16 18

15

34

62

43

aa

aa

2 6

3 4

a

a

62

43

a

a

EXAMPLE 4 Simplifying Complex Fractions (Method 2)


2 2

2 2

2 3

4 1a b ab

a b ab

Solution:

2 2

2

2 2

2 22

2 3

4 1a b ab

a b ab

a b

a b

2 3

4

b a

ab



1 21

41

x x

x

2

2

2 34

4 916

xxxx

2 34

2 3 2 3

4

44

4

44

x x

x x

xx

x x

x x

4

2 3

2 3

2 3

x

x

x

x

4

2 3

x

x

21

41

11

1

x x

xx

x x

x

1 2

4

x x

x

3 1

4

x

x

Remember the same answer is obtained regardless of whether Method 1 or Method 2 is used. Some students prefer one method over the other.

EXAMPLE 6 Deciding on a Method and Simplifying Complex Fractions

Solution:

the fundamental property of rational expressions find the numerical value of a rational expression....

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