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.

2x

3

2x

5+

Add numerators and simplify. 2x

3

2x

5+ =

3 5+

2x=

2x

8 ¿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.1

2x 4–1

3x 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–x

x 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.

3

1

4

3 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

∙22

3 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

+3

𝑥+5

Checkpoint Add and Subtract with Unlike Denominators

Perform the indicated operation.

2x3

7.x 2

2+

ANSWER

2x 2

+3x 4

Checkpoint Add and Subtract with Unlike Denominators

8.3

x 1–x

x 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:

Checkpoint Add and Subtract with Like Denominators

Perform the indicated operation.

1. 3x

2

3x

4+ ANSWER

x2

22.

x 2–

x

x 2–– ANSWER –1

x

2x

4+3.

x 4+–

x 3–ANSWER

x 4+

x 3+

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

Ex:

Example 1 Subtract with Like Denominators

Subtract.

x

x

3+ x

2

3+–

Subtract numerators. x

x

3+ x

2

3+– =

x 3+

x 2–

Checkpoint Extra Examples

21.

x 2–

x

x 2––

ANSWER

–1

x

2x

4+2.

x 4+–

x 3–

x 4+

x 3+

Example 2 Subtract with Unlike Denominators

Subtract .2x 2 –

–1

x 2

– x – 2 4

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.

Example 2 Subtract with Unlike Denominators

Subtract .2x 2 –

–1

x 2

– x – 2 4

SOLUTION

The factors of are and and the factors of are and . So, the LCD is .

x 2

– x – 2 ( )x – 2 ( )x + 1x 2 – 4 ( )x – 2 ( )x + 2

( )x + 2( )x – 2 ( )x + 1

Example 2 Subtract with Unlike Denominators

Rewrite using LCD.

= – 2( )x + 2( )x – 2 ( )x + 1 ( )x – 2( )x + 2 ( )x + 2

( )x + 1

=Subtract numerators.

x + 2 – 2( )x + 1( )x – 2 ( )x + 1 ( )x + 2

= Distributive property

x + 2 – 2x( )x – 2 ( )x + 1 ( )x + 2

– 2

Combine like terms.=

x( )x – 2 ( )x + 1 ( )x + 2

–

( )x + 1

2x 2 –

–1

x 2

– x – 2 4

=1 2

– Factor denominators.( )x – 2 ( )x + 1 ( )x – 2 ( )x + 2

9.5 p.497 # 11, 14, 29, 31, 32, 38,40, 41

Homework: