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Warm up

Warm up. Multiply. Divide. 9.5: Adding and subtracting rational functions. Objectives: 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.

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Warm up

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  1. Warm up Multiply Divide

  2. 9.5: Adding and subtracting rational functions Objectives: You will be able to… Add and subtract rational expressions

  3. 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.

  4. Finding the least common denominator • When our denominators were 3 and 6, finding the LCD was easy. • 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 7 is prime so there are no factors

  5. 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!

  6. Finding LCD with rational functions Break each denominator down into its factors: LCD= or

  7. Find the LCD • ;

  8. Find the LCD • ;

  9. Your turn! Find the LCD

  10. Your turn! Find the LCD LCD: LCD: LCD: LCD:

  11. Warm up • Find the LCD of the following rational expressions:

  12. 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.

  13. Adding and subtracting rational Expressions: Different denominators • LCD: • We want: • Find LCD • Write each with the LCD by multiplying the numerator and denominator of each by the factors that were missing. • Subtract the fractions, leaving the denominator the same

  14. Adding and subtracting rational Expressions: Different denominators Factor denominators: Find LCD: Write both with LCDby multiplying the numerator and denominator of each by what they need. (Remember to distribute!) Add This one needs a 2x +1 This one needs another x

  15. General steps for Adding/Subtracting Rational Expressions • Find the least common denominator! You might need to factor each denominator first… • 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! • Simplify each numerator (FOIL, distribute, combine like terms, etc). • Add or subtract the numerators. • Factor the numerator to simplify, if possible.

  16. Subtract.

  17. More Examples 3) 1) 4) 2) 5)

  18. Complex Fractions

  19. Complex Fractions • A complex fraction is a fraction whose numerator and/or denominator contains fractions. • Ex:

  20. 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!

  21. 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)

  22. Now with expressions • 1. Define your big fraction bar. • 2. Rewrite top fraction. • 3. Flip bottom fraction to multiply by the reciprocal. • 4. Simplify

  23. 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).

  24. 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.

  25. Your turn: Example 3 • Hint #1: Focus on just the top to start • Hint #2: Write the 8 as a fraction over 1

  26. Example 4 • Remember, work on the top and bottom separately, then combine to divide.

  27. More practice

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