9.5 Adding and Subtracting Rational Expressions

4/23/2014

Same Denominator:

When adding or subtracting fractions with same denominators, add/subtract the numerators and

keep the same denominator

Example 1 Add and Subtract with Like Denominators

Perform the indicated operation.

2x3

2x5

+

Add numerators and simplify. 2x

32x5

+ =3 5+

2x=

2x8 ¿

4 ∙22 ∙𝑥 ¿

𝟒𝒙

Different Denominators:

If the denominators are different, find the least common denominator (LCD).

Example 2 Find the LCD of Rational Expressions

Find the least common denominator.

a. 6x1 ,

8x 3

1

List the multiples of the number:6: 6, 12, 18, 24, 30, 36….8: 8, 16, 24, 32…Take the Least Common Multiple (LCM): 24As for the variable, take the one with highest exponent: LCD:

Example 2 Find the LCD of Rational Expressions

Find the least common denominator.

b. 12x 4–

13x 9+

,

2x - 4 = 2(x – 2)3x + 9 = 3(x + 3)2: 2, 4, 6, 8…3: 3, 6, 9, 12…LCD: 6(x-2)(x+3) (Take the LCM and multiply by the rest of their factors)

Example 2 Find the LCD of Rational Expressions

Find the least common denominator.

Factors

LCD:

c. 3 ,x 2–

xx 2 4–

Example 2 Find the LCD of Rational Expressions

Find the least common denominator.

Factors

LCD:

x 1+x 1–,x

2x 1+d.

Basic Example of Adding Fractions with different denominators.

31

43

4: 4, 8, 12, 16..3: 3, 6, 9, 12…LCD: 12

33 ∙ ∙ 44

+¿12

= ❑12

¿9 4 13

Example 3 Add with Unlike Denominators

Perform the indicated operation.

a. 2x3 + x 2

4

Find LCD: 2 𝑥2 ∙ 𝑥𝑥 ∙ 223 𝑥+8

Example 3 Add with Unlike Denominators

Find LCD: (𝑥−2)(𝑥+5) ∙ (𝑥+5)

(𝑥+5) ∙ (𝑥−2)(𝑥−2)

2 𝑥2+10 𝑥+3 𝑥−6

¿2𝑥2+13 𝑥−6(𝑥−2)(𝑥+5)

𝑏 . 2𝑥𝑥−2+

3𝑥+5

Checkpoint Add and Subtract with Unlike Denominators

Perform the indicated operation.

2x37.

x 2

2+

ANSWER

2x 2+3x 4

Checkpoint Add and Subtract with Unlike Denominators

8. 3x 1–

xx 1++

ANSWER

x 2 + 2x + 3( )x – 1 ( )x + 1

Perform the indicated operation.

Example 4 Add with Unlike Denominators

Find LCD: (𝑥+6)(𝑥−6)

∙ (𝑥−6)(𝑥−6)

2 𝑥+3 𝑥−18

¿5 𝑥−18

(𝑥−6)(𝑥+6)

Add:

9.5 p.497 #10, 12, 13, 15, 16-24even, 28, 30, 33, 37, 39

Homework:

9.5 cont’d….Subtracting Rational Expressions

4/25/2014

Example 1 Subtract with Like Denominators

Subtract.

xx

3+ x2

3+–

Subtract numerators. xx

3+ x2

3+– =

x 3+x 2–

Example 2 Subtract with Like Denominators

Perform the indicated operation.

2x 2–

xx 2–

– = = =

Subtracting PolynomialsSwitch the signs of everything that’s being subtracted and combine like terms.Ex:

Example 3 Subtract with Like Denominators

Perform the indicated operation.

= = =x

2x4+ x 4+

– x 3–

Example 4 Subtract with Unlike Denominators

Subtract

LCD:

= 𝑥2 (𝑥+3 )−3 𝑥 (𝑥−2)

∙ 𝑥+3𝑥+3

=

=

∙ 𝑥−2𝑥−2

𝑥2

𝑥−2− 3 𝑥𝑥+3

=

Example 5 Subtract with Unlike Denominators

Subtract 2x 2 ––1

x 2

– x – 2 4

-2

-1

1 -2

=

LCD:

= 1 (𝑥+2 )−2(𝑥+1)

∙ 𝑥+2𝑥+2

=

=

∙ 𝑥+1𝑥+1

Checkpoint Add and Subtract with Unlike Denominators

1 1–x 2

– 1 x + 1

ANSWER

– x – 2( )x – 1( )x + 1

Perform the indicated operation.

9.5 p.497 # 11, 14, 17-23odd, 29, 31, 32, 38,40, 41

Homework: