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Adding and Subtracting Rational Functions: A Comprehensive Guide

This lesson focuses on the addition and subtraction of rational expressions, covering both same and different denominators. Learn how to find the least common denominator (LCD) through factorization, simplify fractions, and apply these concepts in complex fractions. Step-by-step examples will illustrate the processes, ensuring a clear understanding of each stage. By the end, you will be equipped to confidently add and subtract rational expressions and handle complex fractions effectively.

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Adding and Subtracting Rational Functions: A Comprehensive Guide

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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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