1. Rational Functions

2. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3

3. Rational Functions A RATIONAL FUNCTION has the form where p(x) and q(x) are polynomials and q(x) ≠ 0. A rational expression is in SIMPLIFIED FORM if its numerator and denominator have no common factors other than 1.

4. Simplifying Rational Expressions To simplify a rational expression: Let a, b, and c be expressions with b ≠ 0 and c ≠ 0. Then the following property applies: Ex:

5. Simplifying Rational Expressions What if we have an example where we do not have any obvious common factors? Example 1:

6. Simplifying Rational Expressions 0 Example 2: 0 Example 3: 0 Example 4:

7. Simplifying Rational Expressions Classwork – Worksheet Homework – page 577 #3 – 17

8. Do Now Simplify the following rational expressions: 1. 2. 3.

9. Multiplying Fractions 0 How do we multiply fractions? 0 Example: 0 The rule for multiplying rational expressions is the same as multiplying numerical fractions: 0 Multiply numerators 0 Multiply denominators 0 Write new fraction in simplified form. 0 Property:

10. Multiplying Rational Expressions 0 Example 1: Perform the indicated operation. 0 Example 2: Perform the indicated operation.

11. Multiplying Rational Expressions 0 Example 3: Perform the indicated operation. 0 Example 4: Perform the indicated operation.

12. Multiplying Rational Expressions 0 Classwork – Worksheet on Multiplying Rational Expresssions 0 Homework – Textbook page 578 #18-20, 24-33

13. Do Now 0 Perform the indicated operation. Be sure your answer is in simplest form. 1. 2. 3.

14. Dividing Rational Expressions 0 How do we divide fractions? Example: 0 RULE: 0 Keep – change – flip (keep the first fraction the same, change division to multiplication, flip the second fraction) 0 Multiply as usual 0 Property: 0 Excluded values: values that make the denominator equal to zero

15. Dividing Rational Expressions 0 Example 1: Perform the indicated operation and state the excluded values

16. Example 2: Perform the indicated operation and state the excluded values

17. Example 3: Perform the indicated operation and state the excluded values

18. Example 4: Perform the indicated operation and state the excluded values

19. Example 5: Perform the indicated operation and state the excluded values

20. Dividing Rational Expressions 0 Class work: Work in partners on Dividing Rational Expressions Worksheet 0 Homework: Textbook page 578 #34-43

21. Do Now 0 Perform the indicated operation: 1. 2. 3.

22. Adding and Subtracting Rational Expressions -Combine the numerators together -Put the sum or difference in step 1 over the common denominator -Reduce to lowest terms

23. Example:

25. What if we do NOT have an LCD??? Step 1: Find LCD Step 2: Multiply the numerator(s) by the factor that is missing Step 3: Combine and simplify as usual

26. Example:

27. Find the LCDs of the following expressions: a) b) c)

28. DAY 2 Example: Find the LCD: xy Multiply by what’s missing Simplify!

29. Example: Find the LCD: 90xy 2 Multiply by what’s missing Simplify!

30. Example: Find the LCD: -What can’t x equal?? Multiply by what’s missing Simplify!

31. Example: -LCD? -What can’t x equal??

32. Solving Rational Equations Example: **this is a proportion, can solve using cross multiplication 5 (x – 1) = 2 (15) 5x – 5 = 30 5x = 35 x = 7

33. Example: Solve the equation Step 1: Find LCD and state the excluded values 5x ( x + 2 ) x cannot equal 0 or -2 Step 2: Multiply all expressions by LCD to cancel out the denominators Step 3: Simplify, solve, and check

34. Solve: Step 1: Find LCD and state the excluded values Step 2: Multiply all expressions by LCD to cancel out the denominators Step 3: Simplify, solve, and check

35. Solve: