1. 11.3 – Rational Functions

2. Rational functions –

3. Rational functions – algebraic fraction

4. Rational functions – algebraic fraction; numerator and (esp.) denominator are polynomials

5. Ex. Simplify. a. 35yz 2 14y 2 z

6. Rational functions – algebraic fraction; numerator and (esp.) denominator are polynomials Ex. Simplify. a. 35yz 2 14y 2 z 7 · 5 · y · z · z

7. Rational functions – algebraic fraction; numerator and (esp.) denominator are polynomials Ex. Simplify. a. 35yz 2 14y 2 z 7 · 5 · y · z · z 7 · 2 · y · y · z

8. Rational functions – algebraic fraction; numerator and (esp.) denominator are polynomials Ex. Simplify. a. 35yz 2 14y 2 z 7 · 5 · y · z · z 7 · 2 · y · y · z

9. Rational functions – algebraic fraction; numerator and (esp.) denominator are polynomials Ex. Simplify. a. 35yz 2 14y 2 z 7 · 5 · y · z · z 7 · 2 · y · y · z

10. Rational functions – algebraic fraction; numerator and (esp.) denominator are polynomials Ex. Simplify. a. 35yz 2 14y 2 z 7 · 5 · y · z · z 7 · 2 · y · y · z 5z 2y

11. b. x + 4 x 2 + 6x + 8

12. b. x + 4 x 2 + 6x + 8 x + 4 x + 4

13. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4)

14. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4)

15. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4)

16. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4)

17. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 x + 2 x + 2

18. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 ; x ≠ -2 1 ; x ≠ -2 x + 2 x + 2

19. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 ; x ≠ -2, x ≠ -4 1 ; x ≠ -2, x ≠ -4 x + 2 x + 2

20. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 ; x ≠ -2, x ≠ -4 1 ; x ≠ -2, x ≠ -4 x + 2 x + 2 c. x 2 – x – 20 x 2 + 6x + 8

21. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 ; x ≠ -2, x ≠ -4 1 ; x ≠ -2, x ≠ -4 x + 2 x + 2 c. x 2 – x – 20 x 2 + 6x + 8 (x – 5)(x + 4) (x – 5)(x + 4)

22. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 ; x ≠ -2, x ≠ -4 1 ; x ≠ -2, x ≠ -4 x + 2 x + 2 c. x 2 – x – 20 x 2 + 6x + 8 (x – 5)(x + 4) (x – 5)(x + 4) (x + 2)(x + 4) (x + 2)(x + 4)

23. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 ; x ≠ -2, x ≠ -4 1 ; x ≠ -2, x ≠ -4 x + 2 x + 2 c. x 2 – x – 20 x 2 + 6x + 8 (x – 5)(x + 4) (x – 5)(x + 4) (x + 2)(x + 4) (x + 2)(x + 4)

24. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 ; x ≠ -2, x ≠ -4 1 ; x ≠ -2, x ≠ -4 x + 2 x + 2 c. x 2 – x – 20 x 2 + 6x + 8 (x – 5)(x + 4) (x – 5)(x + 4) (x + 2)(x + 4) (x + 2)(x + 4)

25. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 ; x ≠ -2, x ≠ -4 1 ; x ≠ -2, x ≠ -4 x + 2 x + 2 c. x 2 – x – 20 x 2 + 6x + 8 (x – 5)(x + 4) (x – 5)(x + 4) (x + 2)(x + 4) (x + 2)(x + 4)

26. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 ; x ≠ -2, x ≠ -4 1 ; x ≠ -2, x ≠ -4 x + 2 x + 2 c. x 2 – x – 20 x 2 + 6x + 8 (x – 5)(x + 4) (x – 5)(x + 4) (x + 2)(x + 4) (x + 2)(x + 4) x – 5 ; x ≠ -2 x + 2

27. b. x + 4 x 2 + 6x + 8 x + 4 x + 4 (x + 2)(x + 4) 1 ; x ≠ -2, x ≠ -4 1 ; x ≠ -2, x ≠ -4 x + 2 x + 2 c. x 2 – x – 20 x 2 + 6x + 8 (x – 5)(x + 4) (x – 5)(x + 4) (x + 2)(x + 4) (x + 2)(x + 4) x – 5 ; x ≠ -2, x ≠ -4 x + 2