2. ZEROS=ROOTS=SOLUTIONS Equals x intercepts

3. Long Division 1. What do I multiply first term of divisor by to get first term of dividend? 2. Multiply entire divisor by answer to step 1. 3. Subtract result of step 2(distribute the negative) 4. Bring down next term. 5. Start with step 1.Repeat

4. #5 Dividing a polynomial by a polynomial (Long Division) Check

5. POLYNOMIALS – DIVIDING EX – Long division (5x³ -13x² +10x -8) / (x-2) 5x³ - 13x² + 10x - 8x - 2 5x² 5x³ - 10x²-() -3x²+ 10x - 3x -3x² + 6x - () 4x - 8 4x - 8 - () + 4 0 R 0

6. #6 Dividing a polynomial by a polynomial (Long Division) Check

8. f(x) = x + 2 LinearFunction Degree = 1 Maximum Number of Zeros: 1 Polynomial Functions

9. f(x) = x 2 + 3x + 2 QuadraticFunction Degree = 2 Maximum Number of Zeros: 2 Polynomial Functions

10. f(x) = x 3 + 4x 2 + 2 Cubic Function Degree = 3 Maximum Number of Zeros: 3 Polynomial Functions

11. Quartic Function Degree = 4 Maximum Number of Zeros: 4 Polynomial Functions

12. EXAMPLE: ODD A function is odd if the degree which is greatest is odd and even if the degree which is greatest is even Example: even

13. End Behavior Behavior of the graph as x approaches positive infinity (+∞) or negative infinity (- ∞) The expression x→+∞ : as x approaches positive infinity The expression x→-∞ : as x approaches negative infinity

14. End Behavior of Graphs of Linear Equations f(x)→+∞ as x→+∞ f(x)→-∞ as x→-∞ f(x) = x f(x)→-∞ as x→+∞ f(x)→+∞ as x→-∞ f(x) = -x

15. End Behavior of Graphs of Quadratic Equations f(x)→+∞ as x→+∞ f(x)→+∞ as x→-∞ f(x) = x² f(x)→-∞ as x→+∞ f(x)→-∞ as x→-∞ f(x) = -x²

16. End Behavior… Four Possibilities Up on both ends Up on both ends Down on both ends Down on both ends Up on the right & Down on the left Up on the right & Down on the left Up on the left & Down on the right Up on the left & Down on the right

19. End Behavior… Four Prototypes: Up on both ends… y = x 2 Up on both ends… y = x 2 Down on both ends… y = -x 2 Down on both ends… y = -x 2 Up on the right & Down on the left… y = x 3 Up on the right & Down on the left… y = x 3 Up on the left & Down on the right… y = -x 3 Up on the left & Down on the right… y = -x 3

20. End Behavior… Notation: Up on both ends… Up on both ends… Down on both ends… Down on both ends… Up on the right & Down on the left… Up on the right & Down on the left… Up on the left & Down on the right… Up on the left & Down on the right…

21. Investigating Graphs of Polynomial Functions 1.Use a Graphing Calculator to graph each function then analyze the functions end behavior by filling in this statement: f(x)→__∞ as x→+∞ and f(x)→__∞ as x→-∞ a. f(x) = x³c. f(x) = x 4 e. f(x) = x 5 g. f(x) = x 6 b. f(x) = -x³ d. f(x) = -x 4 f. f(x) = -x 5 h. f(x) = -x 6

22. Investigating Graphs of Polynomial Functions How does the sign of the leading coefficient affect the behavior of the polynomial function graph as x→+∞? How is the behavior of a polynomial functions graph as x→+∞ related to its behavior as x→-∞ when the functions degree is odd? When it is even?

23. End Behavior for Polynomial Functions For the graph of If a n >0 and n even, then f(x)→+∞ as x→+∞ and f(x)→+∞ as x→-∞ If a n >0 and n odd, then f(x)→+∞ as x→+∞ and f(x)→-∞ as x→-∞ If a n <0 and n even, then f(x)→-∞ as x→+∞ and f(x)→-∞ as x→-∞ If a n <0 and n odd, then f(x)→-∞ as x→+∞ and f(x)→+∞ as x→-∞

24. 24 Using the Leading Coefficient to Describe End Behavior: Degree is EVEN If the degree of the polynomial is even and the leading coefficient is positive, both ends ______________. If the degree of the polynomial is even and the leading coefficient is negative, both ends ________________.

25. 25 Using the Leading Coefficient to Describe End Behavior: Degree is ODD If the degree of the polynomial is odd and the leading coefficient is positive, the graph falls to the __________ and rises to the ______________. If the degree of the polynomial is odd and the leading coefficient is negative, the graph rises to the _________ and falls to the _______________.

27. Graphing Polynomial Functions f(x)= -x 4 – 2x³ + 2x² + 4x x-3-20123 f(x)

28. Determining End Behavior Match each function with its graph. A. B. C. D.

29. 29 For example, there are an infinite number of polynomials of degree 3 whose zeros are -4, -2, and 3. They can be expressed in the form: Many correct answers

30. POLYNOMIALS – DIVIDING EX – Long division (5x³ -13x² +10x -8) / (x-2) 5x³ - 13x² + 10x - 8x - 2 5x² 5x³ - 10x²-() -3x²+ 10x - 3x -3x² + 6x - () 4x - 8 4x - 8 - () + 4 0 R 0

31. #7 Dividing a polynomial by a polynomial (Long Division)