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POLYNOMIALS – DIVIDING EX – Long division

POLYNOMIALS – DIVIDING EX – Long division. ( 4 x³ -15x² +11x -6) / (x-3). - 3x. + 2. 4 x². R 0. x - 3. 4x³ - 15x² + 11x - 6. -. (. 4 x³ - 12x². ). -3x². + 11x. -. (. -3x² + 9x. ). 2 x. - 6. -. (. 2x - 6. ). 0.

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POLYNOMIALS – DIVIDING EX – Long division

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  1. POLYNOMIALS – DIVIDINGEX – Long division (4x³ -15x² +11x -6) / (x-3) - 3x + 2 4x² R 0 x - 3 4x³ - 15x² + 11x - 6 - ( 4x³ - 12x² ) -3x² +11x - ( -3x² + 9x ) 2x - 6 - ( 2x - 6 ) 0

  2. A function is odd if the degree which is greatest is odd and even if the degree which is greatest is even Example: even Example: odd

  3. 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

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

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

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

  7. End Behavior… Four Prototypes: • Up on both ends… y = x2 • Down on both ends… y = -x2 • Up on the right & Down on the left… y = x3 • Up on the left & Down on the right… y = -x3

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

  9. Investigating Graphs of Polynomial Functions • Use a Graphing Calculator to grph 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) = x³ c. f(x) = x4 e. f(x) = x5 g. f(x) = x6 b. f(x) = -x³ d. f(x) = -x4 f. f(x) = -x5 h. f(x) = -x6

  10. 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?

  11. 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 ________________.

  12. 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 _______________.

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

  14. When you transform a function Inside the parentheses translates left and right Outside the parentheses translates up and down

  15. Transforming Cubic functions

  16. For You to do

  17. Graphing Polynomial Functions f(x)= -x4 –2x³ + 2x² + 4x

  18. Many correct answers 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:

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