Rational Expressions

1. Rational Expressions • Multiplying/Dividing • Adding/Subtracting • Complex Fractions

2. Multiplying / Dividing • Let’s first review how we multiply and divide ordinary fractions. • Do we need a common denominator? No!

3. Multiplying / Dividing • How do we multiply ordinary fractions? • Multiply across: numerators times numerators and denominators times denominators.

4. Multiplying / Dividing • How do we multiply rational expressions with variables? • Multiply across: numerators times numerators and denominators times denominators.

5. Multiplying / Dividing • For Example: Multiply Across Using Power Rules!

6. Multiplying / Dividing • When do we cancel things in fractions? • We can only cancel identicalfactors that appear in both the numerator and denominator. • (x - 3) can only be cancelled by (x - 3), not by x, not by 3.

7. 3 Multiplying / Dividing Here, 3, 9, (x+5) and (x + 5) are all identical factors and can be cancelled. (2x-7) and (7x-2) are factors, but they aren’t identical; we can’t cancel any part of them!

8. Multiplying / Dividing • To simplify: • We must factor first, then we can cancel:

9. Multiplying / Dividing • Simplify: factor, then cancel

10. Multiplying / Dividing • When multiplying rational expressions • factor each numerator and denominator first • then cancel identical factors • then multiply across: numerators by numerators and denominators by denominators.

11. Multiplying / Dividing • Practice:

12. = = Multiplying / Dividing • Solution: =

13. Multiplying / Dividing • Now on to dividing. • This is exactly like multiplying, except for ONE step. • We multiply by the reciprocal of the 2nd fraction!

14. Multiplying / Dividing • Divide: • Change it to multiplication and flip the 2nd fraction:

15. = = Multiplying / Dividing • Divide: Now proceed like a multiplication problem. Factor first, cancel, multiply.

16. Adding/Subtracting • What do we have to do to add or subtract ordinary fractions? • Change one or both fractions so they have the same common denominator.

17. Adding/Subtracting • Find the LCD for two fractions with monomial denominators: • The key is that the LCD be something we can reach by multiplyingeach denominator by missing terms.

18. Same den. (LCD) Adding/Subtracting • If we multiply the 1st denominator by 3a we get: • If we multiply the 2nd denominator by 5b we get:

19. Adding/Subtracting • Once we have the same denominator, we add the numerators: • After adding the numerators, try to factor and cancel in the final fraction if possible.

20. Adding/Subtracting • Find the LCD for two fractions with polynomial denominators: • First we must factor the denominators...

21. Adding/Subtracting • The LCD will need to include at least : • One (x+2) factor from the 1st fraction • One (x+3) factor from the 1st fraction • Two (x+2) factors from the 2nd fraction We don’t need three (x+2) terms, two will satisfy the needs of BOTH fractions!

22. (x+2) (x+3) (x+2) (x+3) Adding/Subtracting • Get the LCD: (x+2)(x+2)(x+3)

23. = Adding/Subtracting • Subtract the numerators: • We cannot factor the numerator, so we are finished (don’t try to cancel anything).

24. Adding/Subtracting • Practice:

25. Adding/Subtracting • Practice: Factor: LCD must contain at least: a multiple of 2, a multiple of 4, a factor of (x-3). 2* 2*

26. Adding/Subtracting Subtract: Here, we do have factors to cancel:

27. Complex Fractions • Complex fractions are those fractions whose numerators &/or denominators contain fractions. • To simplify them, we just multiply the top & bottom by the LCD.

28. Complex Fractions • Example What would the LCD be? The denominators are 3 and 6, the LCD is 6.

29. 2 *6 *6 Complex Fractions • Multiply the top & bottom both by 6:

30. Complex Fractions • Simplify:

31. Complex Fractions • What would the LCD be? The denominators are y and x, the LCD is xy.

32. Complex Fractions • Multiply top & bottom by LCD:

33. Complex Fractions • The final answer is: • We cannot cancel any terms in this fraction!