Rational Expressions Finding LCD

Algebra

1

5

2

5

Review: Adding FractionsYou can only add like to like

Same Denominators

1

5

2

5

3

5

Different DenominatorsMake them the same

find a common denominator

The product of all denominators is always a common denominatorBut not always the least common denominator

Finding the LCD

Example:

1

12

4

15

Factor the Denominators

15 3 5 12 2 2 3

Assemble LCD

15 3 5 12 2 2 3

2 2 3 5 60

Each factor in the denominators needs to be used at least once in the LCD.

Build up Fractions to LCD

1

12

4

15

×5

×5

×4

×41

12

4

15

5

60

16

60

Add Numerators

5

60

16

60

5

60

16

60

21

60

5

60

16

60

21

60

7

20

And Reduce if Needed

12

2

6

2 nn

12

2

26

22 nn

12

2

12

4 nn

12

6n

2

n

Rational Expressions

Example:

x

x

x

x x

1

1

2

2 12 2

Factor the Denominators

x x x2 1 1 1 ( )( )

x xx x

2 2 11 1

( )( )

Assemble LCD

( )( )x x 1 1

( )( )x x 1 1

( )( )( )x x x 1 1 1

DE

NO

MIN

AT

OR

S

Each factor in the denominators needs to be used at least once in the LCD.

Build up Fractions to LCD

x

x x

x

x x

1

1 1

2

1 1( )( ) ( )( )

LCD x x x ( )( )( )1 1 1

x ( )1

x ( )1)( x( )1

x( )1FACTORED

DENOMINATORS

Add Numerators

( )( ) ( )

( )( )( )

x x x x

x x x

1 1 2 1

1 1 1

x x x x

x x x

2 22 1 2 2

1 1 1

( )( )( )

Simplify Numerator

3 1

1 1 1

2x

x x x

( )( )( )

( )( ) ( )

( )( )( )

x x x x

x x x

1 1 2 1

1 1 1

Summary of Steps

1. Factor denominators

2. Assemble LCD

3. Build up Fractions

4. Add/Subtract Numerators

5. Simplify Numerator

6. Reduce Factors if Possible

Add or subtract as indicated.

2

3

9 2 6

x

x x

3

( 3)( 3) 2( 3)

x

x x x

So the LCD is 2(x + 3)(x – 3)

3

2( 3)( 3) 2( 3)( 3)

x

x x x x

(2) ( 3)x 2 3( 3)

2( 3)( 3)

x x

x x

2 3 9

2( 3)( 3)

x x

x x

5 9

2( 3)( 3)

x

x x

Subtract So the LCD is (x – 2)(x + 3)(x – 3)

2 2

6

9 5 6

x x

x x x

6

( 3)( 3) ( 2)( 3)

x x

x x x x

6

( 2)( 3)( 3) ( 2)( 3)( 3)

x x

x x x x x x

( 2)x ( 3)x

6 ( 2) ( 3)

( 2)( 3)( 3)

x x x x

x x x

2 26 12 3

( 2)( 3)( 3)

x x x x

x x x

25 15

( 2)( 3)( 3)

x x

x x x

5 ( 3)

( 2)( 3)( 3)

x x

x x x

5

( 2)( 3)

x

x x

Rational Expressions Finding LCD

Algebra 1 &

Algebra 1 Honors