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)

34

23

LCD is 12.

Find equivalentfractions using the LCD.

9

12

812

=9 + 812

=1712

Collect the numerators, keeping the LCD.

Adding Fractions - A Review

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

denominator!

51

53 . a

54

21

32 . b

63

64

61

43

21

.c34

21

64

32

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

156x

2

3x22

52x

33

Simplify...

23x

52x

Find the LCD: 6x

Now, rewrite the expression using the LCD of 6x

Add the fractions...4 15

6x

= 19 6x

65m

8

3m2n

7mn2

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

27

23 .

x24

x2

46

43 .

xxxb

463

xx

4)2(3

xxor

Example 2

3x 23

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 14

3x 1

2

5x 33

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

12

6x 3 18x 6 20x 12

12

8x 21

12

Example 4 Simplify:

4a3b

2b3a

LCD = 3ab

3ab

4a2 2b2

3ab

a ≠ 0b ≠ 0

Example 5

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

Simplify:

Adding and Subtracting with polynomials as denominators

Simplify:

3x 6

x 2 x 2

8x 16x 2 x 2

3

(x 2)x 2x 2

8(x 2)

x 2x 2

Simplify...

3x 2

8x 2

Find 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)

2x 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 3634

xxx

x ** Needs a common denominator 1st!

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

your LCD.)12(334

23

xxx

xLCD: 3x3(2x+1)

)12(3)12(3)12(4

3

2

3

xxx

xxx

)12(3)12(4

3

2

xx

xx)12(348

3

2

xxxx

Example 6 Simplify:

91

961

22

xxxx

)3)(3(1

)3)(3(1

xxxx

x

)3()3()3(

)3()3()3)(1(

22

xx

xxxxx

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

)3()3()3()3)(1(

2

xxxxx

)3()3(333

2

2

xx

xxxx)3()3(

632

2

xxxx

Example 7 Simplify:

2xx 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:

3xx2 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

4xx2 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:

x 3x2 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

Pg 173 # 1 – 21 odd