6.4 Adding and Subtracting Rational Expressions

Objective 1

Add rational expressions having the same denominator.

Slide 6.4-3

Add rational expressions having the same denominator.

We find the sum of two rational expressions with the same procedure that we used in Section 1.1 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

Slide 6.4-4

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

Slide 6.4-5

Adding Rational Expressions (Same Denominator)CLASSROOM EXAMPLE 1

Objective 2

Add rational expressions having different denominators.

Slide 6.4-6

We use the following steps, which are the same as those used in Section 1.1 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.

Slide 6.4-7

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

Slide 6.4-8

Adding Rational Expressions (Different Denominators)CLASSROOM EXAMPLE 2

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

Slide 6.4-9

Adding Rational ExpressionsCLASSROOM EXAMPLE 3

Solution:

2 2

2 3

5 4 1

k

k k k

2 3

4 1 1 1

k

k k k k

Add. Write the answer in lowest terms.

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

Slide 6.4-10

Adding Rational ExpressionsCLASSROOM EXAMPLE 4

Add. Write the answer in lowest terms.

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

Slide 6.4-11

Adding Rational Expressions (Denominators Are Opposites)CLASSROOM EXAMPLE 5

Objective 3

Subtract rational expressions.

Slide 6.4-12

Subtract rational expressions.

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

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

Slide 6.4-13

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.

Slide 6.4-14

Subtracting Rational Expressions (Same Denominator)

Subtract. Write the answer in lowest terms.

CLASSROOM EXAMPLE 6

6 1

2 3a a

Subtract. Write the answer in lowest terms.

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

Slide 6.4-15

Subtracting Rational Expressions (Different Denominators)CLASSROOM EXAMPLE 7

4 3 1

1 1

x x

x x

Subtract. Write the answer in lowest terms.

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

Slide 6.4-16

Subtracting Rational Expressions (Denominators Are Opposites)CLASSROOM EXAMPLE 8

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

Slide 6.4-17

Subtracting Rational Expressions

Subtract. Write the answer in lowest terms.

CLASSROOM EXAMPLE 9