Warm up

6

1

8

2

3

1

3

1

5

21

2

11

17

5

3

11

4

12

Adding and Subtracting Rational Expressions

Goal 1 Determine the LCM of polynomials

Goal 2 Add and Subtract Rational Expressions

What is the Least Common Multiple?

                                       Least Common Multiple (LCM) - smallest number or

polynomial into which each of the numbers or polynomials will divide evenly.

Fractions require you to find the Least Common Multiple (LCM) in order to add and subtract them!

The Least Common Denominator is the LCM of the denominators.

Find the LCM of each set of Polynomials

1) 12y2, 6x2 LCM = 12x2y2

2) 16ab3, 5a2b2, 20ac LCM = 80a2b3c

3) x2 – 2x, x2 - 4 LCM = x(x + 2)(x – 2)

4) x2 – x – 20, x2 + 6x + 8

LCM = (x + 4) (x – 5) (x + 2)

3

4

2

3

LCD is 12.

Find equivalentfractions using the LCD.

9

12

8

12

=9 + 8

12

=17

12

Collect the numerators, keeping the LCD.

Adding Fractions - A Review

Remember: When adding or subtracting fractions, you need a common

denominator!

5

1

5

3 . a

5

4

2

1

3

2 . b

6

3

6

4

6

1

43

21

.c3

4

2

1

6

4

3

2

When Multiplying or Dividing Fractions, you don’t need a common Denominator

1. Factor, if necessary.2. Cancel common factors, if possible.

3. Look at the denominator.

4. Reduce, if possible.5. Leave the denominators in factored form.

Steps for Adding and Subtracting Rational Expressions:

If the denominators are the same,

add or subtract the numerators and place the result over the common denominator.

If the denominators are different,

find the LCD. Change the expressions according to the LCD and add or subtract numerators. Place the result

over the common denominator.

Addition and SubtractionIs the denominator the same??

• Example: Simplify

4

6x

15

6x

2

3x

2

2

5

2x

3

3

Simplify...

2

3x

5

2xFind the LCD: 6x

Now, rewrite the expression using the LCD of 6x

Add the fractions...4 15

6x

= 19 6x

6

5m

8

3m2n

7

mn2

15m2n2

18mn2 40n 105m

15m2n2

LCD = 15m2n2

m ≠ 0n ≠ 0

6(3mn2) + 8(5n) - 7(15m)

Multiply by 3mn2

Multiply by 5nMultiply by

15m

Example 1 Simplify:

Examples:

xxa

2

7

2

3 .

x2

4

x

2

4

6

4

3 .

xx

xb

4

63

x

x4

)2(3

x

xor

Example 2

3x 2

3 2x

4x 1

5

15

15x 10 30x 12x 3

15

27x 7

15

LCD = 15

(3x + 2) (5) - (2x)(15) - (4x + 1)(3)

Mult by 5

Mult by 15 Mult by

3

Example 3 Simplify:

2x 1

4

3x 1

2

5x 3

3

3(2x 1) 6(3x 1) 4(5x 3)

12

6x 3 18x 6 20x 12

12

8x 21

12

Example 4 Simplify:

4a

3b

2b

3a

LCD = 3ab

3ab

4a2 2b2

3ab

a ≠ 0b ≠ 0

Example 5

(a) (b)(4a) - (2b)

Simplify:

ab

ba

3

)2(2 22

Adding and Subtracting with polynomials as denominators

Simplify:

3x 6

x 2 x 2

8x 16

x 2 x 2

3

(x 2)

x 2

x 2

8

(x 2)

x 2

x 2

Simplify...

3

x 2

8

x 2Find the LCD:

Rewrite the expression using the LCD of (x + 2)(x – 2)

3x 6 (8x 16)

(x 2)(x 2)

– 5x – 22 (x + 2)(x – 2)

(x + 2)(x – 2)

3x 6 8x 16

(x 2)(x 2)

2

x 3

3

x 1

(x 3)(x 1)

LCD =(x + 3)(x + 1)

x ≠ -1, -3

2x 2 3x 9

(x 3)(x 1) 5x 11

(x 3)(x 1)

2 + 3(x + 1) (x + 3)

Multiply by (x + 1)

Multiply by (x + 3)

Adding and Subtracting with Binomial Denominators

233 363

4

xx

x

x ** Needs a common denominator 1st!

Sometimes it helps to factor the denominators to make it easier to find

your LCD.)12(33

423

xx

x

xLCD: 3x3(2x+1)

)12(3)12(3

)12(43

2

3

xx

x

xx

x

)12(3

)12(43

2

xx

xx)12(3

483

2

xx

xx

Example 6 Simplify:

9

1

96

122

xxx

x

)3)(3(

1

)3)(3(

1

xxxx

x

)3()3(

)3(

)3()3(

)3)(1(22

xx

x

xx

xx

LCD: (x+3)2(x-3)

)3()3(

)3()3)(1(2

xx

xxx

)3()3(

3332

2

xx

xxxx

)3()3(

632

2

xx

xx

Example 7 Simplify:

2x

x 1

3x

x 2

(x 1)(x 2)

x ≠ 1, -2 2x2 4x 3x2 3x

(x 1)(x 2)

x2 7x

(x 1)(x 2)

2x (x + 2) - 3x (x - 1)

Example 8 Simplify:

)2)(1(

)7(

xx

xx

3x

x2 5x 6

2x

x2 2x 3(x + 3)(x + 2) (x + 3)(x - 1)

LCD(x + 3)(x + 2)(x - 1)

(x 3)(x 2)(x 1)3x

3x2 3x 2x2 4x

(x 3)(x 2)(x 1)

x2 7x

(x 3)(x 2)(x 1)

x ≠ -3, -2, 1

- 2x(x - 1) (x + 2)

Simplify:Example 9

)1)(2)(3(

)7(

xxx

xx

4x

x2 5x 6

5x

x2 4x 4(x - 3)(x - 2) (x - 2)(x - 2)

(x 3)(x 2)(x 2) 4x + 5x

4x2 8x 5x2 15x

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

9x2 23x

(x 3)(x 2)(x 2) x ≠ 3, 2

(x - 2) (x - 3)

LCD(x - 3)(x - 2)(x - 2)

Example 10 Simplify:

)2)(2)(3(

)239(

xxx

xx

x 3

x2 1

x 4

x2 3x 2(x - 1)(x + 1) (x - 2)(x - 1)

(x 1)(x 1)(x 2) (x + 3) - (x - 4)

(x2 x 6) (x2 3x 4)

(x 1)(x 1)(x 2)

4x 2

(x 1)(x 1)(x 2) x ≠ 1, -1, 2

(x - 2) (x + 1)

LCD(x - 1)(x + 1)(x - 2)

Simplify:Example 11

)2)(1)(1(

)12(2

xxx

x

Warm up

• Type 2 WAC• Using complete sentences explain how you

would determine the least common denominator for the following expressions.

• Include the correct LCD in your explanation.

22

3,

2

5,

6 xxxx

x