2. 1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. START UP DAY 12 Sketch each graph: 1. A Polynomial function with a degree of 4, a positive leading coefficient & Zeros @ x = 1, -3, 3, 0 1. A Polynomial function with a degree of 5, a negative lead coefficient and Zeros @

3. 2 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Pre Calculus Day 12 Objective: SWBAT use long and synthetic division to determine the zeros of polynomial functions. Essential Question: How can the use of division lead to the understanding of zeros and end behavior help create a function graph? Classwork/Home Learning: p. 205# 1, 4, 9, 15, 23, 26, 39, 43, 53 /p. 205# 5, 12, 17, 22, 55 + MathXL Quiz Review (Lessons 2.1 through 2.5)

4. 3 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 2.4 Real Zeros of Polynomial Functions Demana, Waits, Foley, Kennedy

5. 4 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. What you’ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower Bounds … and why These topics help identify and locate the real zeros of polynomial functions.

6. 5 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Division Algorithm for Polynomials

7. 6 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Using Polynomial Long Division

8. 7 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

9. 8 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Remainder Theorem

10. 9 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Using the Remainder Theorem

11. 10 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

12. 11 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Factor Theorem

13. 12 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Fundamental Connections for Polynomial Functions For a polynomial function f and a real number k the following statements are equivalent: 1. x = k is a solution (or root) of the equation f(x) = 0 2. k is a zero of the function f. 3. k is an x-intercept of the graph of y = f(x). 4. x – k is a factor of f(x).

14. 13 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn: Using Synthetic Division

15. 14 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. The First step in synthetic division is to set up your problem by placing your “x” “x” value on the shelf in the upper left hand box. Then, using the coefficients only Place each term’s coefficient from the highest power term to the lowest, in order, from left to right. Make sure to include a “0” “0” for any missing powers of “x”.“x”. Synthetic Division

16. 15 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Given f(x), find f(2) using synthetic division 2 1 -3-412 Notice the ordering has changed to allow for the polynomial to be written from the highest power to the lowest power of “x”. Drop the first coefficient straight down.

17. 16 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 21-3-412 2 1 The next step is to multiply your “x” “x” value by the bottom line element, place the resultant on the shelf and add vertically. Place your sum under the shelf and repeat this process until your done!

18. 17 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 2 1 -3 -4 12 2 -2 -12 1 -1 -6 0 F(2)=0 Which means that x=2 is a zero of the function, since the remainder is “0”! This is called the “FACTOR THEOREM” (X-2) IS A FACTOR OF f(x)

19. 18 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Try to use synthetic division to evaluate f(1) 1-3-412 1-2-6 1-2-66 This means that f(1)=6 OR (1,6) is a point on your graph This is known as the “REMAINDER THEOREM”

20. 19 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn: Using Synthetic Division

21. 20 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

22. 21 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Rational Zeros Theorem

23. 22 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Finding the Real Zeros of a Polynomial Function

24. 23 Copyright © 2015, 2011, and 2007 Pearson Education, Inc.

25. 24 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Upper and Lower Bound Tests for Real Zeros

26. 25 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Finding the Real Zeros of a Polynomial Function

27. 26 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

28. 27 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Finding the Real Zeros of a Polynomial Function

29. 28 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Finding the Real Zeros of a Polynomial Function

30. 29 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

31. 30 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution continued

32. 31 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution continued