WARM UPMultiply

1)

2)

Divide

3)

4) 2

54

32

3

2

2

2

27

12

3

9

2

34

65

24

y

yx

yx

yx

x

xx

xx

xx

2

94

2

)32( 2

x

x

x

xx

3

116

3

)14( 2

x

x

x

xx

9.5: ADDING AND SUBTRACTING RATIONAL FUNCTIONSObjectives: You will be able to…

• Add and subtract rational expressions

ADDING AND SUBTRACTING FRACTIONS Same denominator: Add

(or subtract) the numerators, keep the denominator the same.

Different denominators: Convert them both to the least common denominator, then add/subtract the numerators, keeping the new denominators the same.

Always reduce if possible at the end.

FINDING THE LEAST COMMON DENOMINATOR

How about in the following example?

First we break the denominators into their prime factors.

Every factor needs to be represented in our common denominator…

…so it needs to have a factor of 7,a factor of 3, and a two factors of 2.

Our common denominator will be 22∙3∙7=84

When our denominators were 3 and 6, finding the LCD was easy.

7 is prime so there are

no factors 12=3 ∙2 ∙2=3 ∙2 2

21=7 ∙3

FINDING THE LEAST COMMON DENOMINATOR

8=23 and 12=22∙3

For factors that are the same with different exponents, take the highest exponent.

LCD=23∙3=24

You try!

FINDING LCD WITH RATIONAL FUNCTIONS

Break each denominator down into its factors:

LCD=

or

233 36;

3

4

xx

x

x

3 ∙ 𝑥33 𝑥2(2𝑥+1)

3 ∙ 𝑥2 ∙(2 𝑥+1)

FIND THE LCD

;

23𝑥+2

;5

𝑥+2

FIND THE LCD

;

2𝑥2

𝑥2+6 𝑥+9;

5 𝑥(𝑥+3)(𝑥−5)

YOUR TURN! FIND THE LCD

YOUR TURN! FIND THE LCD

LCD: LCD:

LCD: LCD:

WARM UP

1. Find the LCD of the following rational expressions:

ADDING AND SUBTRACTING RATIONAL EXPRESSIONS: SAME DENOMINATORS Just like with fractions, if they have the same

denominator already, we can just add or subtract the numerators Make sure to simplify at the end!

Examples:

1.

2.

xx 2

7

2

3

xx

2

2

4

4

6

4

3

xx

x

4

63

x

x

ADDING AND SUBTRACTING RATIONAL EXPRESSIONS: DIFFERENT DENOMINATORS

LCD:

We want: 1. Find LCD

2. Write each with the LCD by multiplying the numerator and denominator of each by the factors that were missing.

3. Subtract the fractions, leaving the denominator the same

❑30𝑥3

− ❑30 𝑥3

     = 8 𝑥−930𝑥3

Factor denominators:

Find LCD:

Write both with LCD by multiplying the numerator and denominator of each by what they need. (Remember to distribute!)

Add

233 363

4

xx

x

x

1233

423

xx

x

x

This one needs a 2x +1 This one needs another x

xx

xx

x

x

x

x

12312

12

3

423

123123

483

2

3

xx

x

xx

x

ADDING AND SUBTRACTING RATIONAL EXPRESSIONS: DIFFERENT DENOMINATORS

3 𝑥3(2𝑥+1)

¿ 𝑥2+8𝑥+43 𝑥3(2𝑥+1)

GENERAL STEPS FOR ADDING/SUBTRACTING RATIONAL EXPRESSIONS1. Find the least common denominator! You

might need to factor each denominator first…

2. Figure out what each fraction is missing and multiply the numerator and denominator of each by the missing piece(s). Leave denominator in factored form!

3. Simplify each numerator (FOIL, distribute, combine like terms, etc).

4. Add or subtract the numerators.

5. Factor the numerator to simplify, if possible.

SUBTRACT. 9

1

96

122

xxx

x

MORE EXAMPLES 3)

1) 4)

2) 5)

COMPLEX FRACTIONS

COMPLEX FRACTIONS

A complex fraction is a fraction whose numerator and/or denominator contains fractions.

Ex:

WOAH, WOAH, WOAH: FRACTIONS IN FRACTIONS?! What does the fraction bar mean?

Division

And what do we do when we divide fractions?

Flip the second fraction and multiply

WE ALREADY KNOW HOW TO DO THIS!

LET’S TRY AN EASY ONE (WITH NUMBERS) =

First, make the middle fraction bar nice and big so you can clearly see the top and bottom fractions.

Rewrite the top fraction as it is

Next, multiply by the reciprocal of the bottom fraction. (Flip the bottom fraction upside down and multiply)

5723

¿1514

∙32

NOW WITH EXPRESSIONS

1. Define your big fraction bar.

2. Rewrite top fraction.

3. Flip bottom fraction to multiply by the reciprocal.

4. Simplify

ADDING AND SUBTRACTING WITHIN COMPLEX FRACTIONS Start by looking at the

numerator and denominator separately.

Follow our steps from previous classes to make the numerator and denominator each one fraction.

Then follow your steps for dividing fractions (flip the bottom and multiply).

4+1𝑥

3+ 2𝑥2

COMPLEX FRACTION STEPS

Step 1: Clearly separate numerator and denominator

Step 2: Add/subtract the numerator (if necessary) by following our previous steps.

Step 3: Add/subtract the denominator (if necessary) by following our previous steps.

Step 4: Write the new numerator over the new denominator.

Step 5: Divide the fractions by flipping the fraction in the denominator and multiplying.

YOUR TURN: EXAMPLE 3

Hint #1: Focus on just the top to start

Hint #2: Write the 8 as a fraction over 1

EXAMPLE 4 Remember, work on the top and

bottom separately, then combine to divide.

MORE PRACTICE