Chapter 9 - Rational Expressions

CHAPTER 9

Simplifying, Multiplying, Dividing, Adding and Subtracting

Radical Functions

Simplifying Rational Expressions• Any expression that has a variable in the

denominator is a rational expression• A rational expression is in simplest form if the

numerator and the denominator have no common factors except 1

• To simplify rational expressions, you will often have to factor (chapter 9)

• Ex1. Simplify 2

3 159 20

xx x

• Simplify each of the following• Ex2.

• Ex3.

2

6 124x

x

2

2

207 12

m mm m

Multiplying and Dividing Rational Expressions

• You multiply rational expressions like you do rational numbers

• Be sure to reduce if possible• Multiply and simplify (if possible). Leave in

factored form.• Ex1. Ex2. 4

3 6x x

x x

2

4 10 155 12

x xx x

• Divide rational expressions just like you would rational numbers

• Leave answers in factored form• Remember to flip the 2nd rational expression• Divide• Ex3. Ex4.

2

2 4 85 3 15

x xx x x

2

22

7 12 5 43

x x x xx x

12 – 5 Dividing Polynomials• To divide a polynomial by a monomial, divide each

term of the polynomial by the monomial divisor (you will often end up with rational parts to your function)

• Ex1. Divide.• To divide a polynomial by a binomial, you follow the

same process you use in long division• If the dividend has terms missing (i.e. x³ + x + 1) you

must include that term (0x² in this case)

4 3 2(6 10 8 ) 2x x x x

• Divide.• Ex2.

• Ex3.

• Ex4.

3 27 5 21 3x x x x

4 3 23 8 7 2 3 2x x x x x

34 5 3 2 1m m m

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 Polynomials1) 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

23x

22

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

3m2n7mn2

15m2n2

18mn2 40n 105m

15m2n2LCD = 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 15

15

15x 10 30x 12x 3

15

27x 715

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 12

5x 33

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

12

6x 3 18x 6 20x 12

12

8x 2112

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

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)

2x 3

3x 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

xx

xxx

x

)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

xxx

xx

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

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

2

xx

xxx

)3()3(333

2

2

xx

xxxx)3()3(

632

2

xx

xx

Example 7 Simplify:

2xx 1

3xx 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

• Add or subtract• Ex1. Ex2.

• Ex3. Ex4.

6 72 2x x

4 1 53 4 3 4

x xx x

2

1 69 14 7

x xx x x

2 1 3

4 2x x

x x