Chapter 3- Polynomial and Rational Functions

1. Chapter 3- Polynomial and Rational Functions 16 Days

2. 3.1 Polynomial Functions of Degree Greater Than 2 One Day

3. Essential Question • How does a mountain range relate to polynomials and their graphs?

4. Intermediate Value Theorem • Definition: If f is a polynomial function and f(a) ≠f(b) for a<b, then f takes on every value between f(a) and f(b) in the interval [a, b]. Facts about IVT: • f is continuous from f(a) to f(b) • If f(a) and f(b) have opposite signs (1 positive and 1 negative), there is at least one number c between a and b such that f(c)=0.

5. Diagram of “IVT”

6. Steps Using IVT • Step 1 • Identify a and b values • Step 2 • Plug a and b into f(x) and solve • Step 3 • If f(a) and f(b) have opposite signs, you know that f(c)=0 for at least one real number between a & b

7. Example of IVT • Using the Intermediate Value Theorem • Show that f(x)= x⁵+2x⁴-6xᶟ+2x-3 has a zero between 1 and 2. • Step 1: identify a and b • a=1, b=2 • Step 2: solve f(a) and f(b) • f(a)= f(1)= 1+2-6+2-3=-4 • f(b)=f(2)=32+32-48+4-3=17 • Step 3: do f(a) and f(b) have opposite signs? • f(a)=-4 and f(b)=17, yes! • Conclusion: • there is a c where f(c)=0 between 1 and 2

8. Sketching Graphs • Sketching graphs of degree greater than 2 • Step 1 • Find the zeros • Step 2 • Create a table showing intervals of positive or negative signs for f(x). • Step 3 • Find where f(x)>0 and where f(x)<0

9. Step 2 Explained • Construct a table showing intervals of positive or negative values of f(x) • This means that f(x)>0 if x is in • This means that f(x)<0 if x is in .

10. Sketching Graphs • Sketching the graph of a polynomial function of degree 3 • Ex • Step 1: Find Zeros • Group terms • Factor out x² and -4 • Factor out (x+1) • Difference of squares • Therefore, the zeros are -2, 2, and -1

11. Sketching Graphs • Step 2: Create a Table

12. Sketch a Graph • f(x)>0 if x is in • f(x)<0 if x is in • My version of graph (there are multiple ways to do this graph) y x

13. Sketching a graph given sign diagram • Ex • Find the intervals • f(x) is below x-axis when x is in • f(x) is above x-axis when x is in • Sketch graph y x

14. Estimating Zeros • Steps • Step 1: Assign f(x) to Y1 on a graphing calculator • Step 2: Set x and y bounds large enough to see from [-15,-15] and [-15,15] • This allows us to gauge where the zeros lie from a broad perspective • Step 3: Readjust bounds once you know where zeros are more likely to be found • Step 4: Use zero or root feature on calculator to estimate the real zero

15. Estimating Zeros • Ex/ Estimate the real zeros of • Graph this function as Y1 in calculator • Set bounds as [-15,15] by [-15,15]. • Readjust to where zeros might exist, you may use [-1,3] by [-1,3] • Find actual root by using zero or foot feature on your calculator Actual root is 0.127

16. Homework • P. 227 #1,5,7,15,16,18,23,25,34,41,50 • If you need help, check with the person beside you.

17. 3.2 Properties of Division Two Days

18. Properties of Division • If g(x) is a factor of f(x), then f(x) is divisible by g(x). • For example, is divisible by each of the following . • Notice that is not divisible by . • However, we can use Polynomial Long Division to find a quotient and a remainder.

19. Let’s review long division… • We will first divide 178 by 8 to find the quotient and remainder.

20. Steps for dividing polynomials • Step 1: Make sure the polynomial is written in descending order. If any terms are missing, use a zero to fill in the missing term (this will help with the spacing). • Step 2: Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol. • Step 3: Multiply (or distribute) the answer obtained in the previous step by the polynomial in front of the division symbol. • Step 4: Subtract and bring down the next term. • Step 5: Repeat Steps 2, 3, and 4 until there are no more terms to bring down. • Step 6: Write the final answer. The term remaining after the last subtract step is the remainder and must be written as a fraction in the final answer. • http://www.youtube.com/watch?v=smsKMWf8ZCs

21. Polynomial Long Division • Find the quotient of and .

22. Long Division Example cont. • Once we have a quotient and remainder, we must write our final answer.. Final Answer:

23. Lets do another.. • Divide by .

24. One More Practice Problem.. • Divide by .

25. Remainder Theorem • If a polynomial f(x) is divided by x-c, the the remainder is f(c). • Example:

26. Factor Theorem • A polynomial f(x) has a factor x-c if and only if f(c)=0. • Example:

27. Finding a Polynomial with Known Zeros • Find a polynomial f(x) of degree 3 that has zeros 3, -1, and 1.

28. Homework • pg 236 (# 1,2,8,17-20)

29. For Extra Examples • http://www.mesacc.edu/~scotz47781/mat120/notes/divide_poly/long_division/long_division.html

30. 3.2 Day 2 – Synthetic Division

31. Synthetic Division • Synthetic Division is the “shortcut” method for dividing polynomials.

32. Long Division Synthetic Division Long vs. SyntheticDivision of Polynomials

33. Lets try a few • Use synthetic division to find the quotient q(x) and remainder r

34. More Synthetic Division • Using synthetic division to find zeros. • What must we show for a value to be a zero of f(x)? Think Factor Theorem…

35. More Synthetic Division • Use synthetic division to find f(3) if

36. Check Point • You should now recognize the following equivalent statements. If f(a)=b, then: • 1. The point (a,b) is on the graph of f. • 2. The value of f at x=a equals b; ie f(a)=b. • 3. If f(x) is divided by x-a, then the remainder is b. • Additionally, if b=0 then the following are also equivalent. • 1. The number a is a zero of the function f. • 2. The point (a,0) is on the graph of f; a is an x-int. • 3. The number a is a solution of the equation f(x)=0. • 4. The binomial x-a is a factor of the polynomial f(x).

37. Homework • pg 237 (# 21-29 odd,32,35,38)

38. 3.3 Zeros of Polynomials Two Days

39. Fundamental Theorem of Algebra • If a polynomial f(x) has positive degree and complex coefficients, then f(x) has at least one complex zero.

40. Complete Factorization Theorem for Polynomials • If f(x) is a polynomial of degree n>0, then there exist n complex numbers such that • Where a is the leading coefficient of f(x). Each number is a zero of f(x).

41. Theorem on the Maximum Number of Zeros of a Polynomial • A polynomial of degree n>0 has at most n different complex zeros.

42. Multiplicities of Zeros f f f

43. Finding Multiplicities • Find the zeros of the polynomial, state the multiplicity of each zero, find the y-int, and sketch the graph.

44. Finding a Polynomial with Prescribed Zeros • Find a polynomial f(x) in factored for that has degree 3; has zeros 3, 1, and -1; and satisfies f(-2)=3.

45. Theorem on the Exact Number of Zeros of a Polynomial (N-zeros Theorem) • If f(x) is a polynomial of degree n>0 and if a zero of multiplicity m is counted m times, then f(x) has precisely n zeros. • Exactly how many zeros exist for the following polynomial?

46. End Behavior of Polynomials

47. Homework • pg 249 (# 1-7 odd, 11-15 odd, 19-23 odd)

48. Descartes’ Rule of Signs • If f(x) is a polynomial with real coefficients and a nonzero constant term, then: • 1. The number of positive real zeros of f(x) either is equal to the number of variations of sign in f(x) or is less than that number by an even integer. • 2. The number of negative real zeros of f(x) either is equal to the number of variations of sign in f(-x) or is less than that number by an even integer.

49. Using Descartes’ Rule of Signs • We can use the N-Zeros Thm to determine the total number of zeros possible. • Use Descartes’ Rule of Signs to determine the total possible number of positive and negative zeros (and all lesser combinations). • Any unaccounted for zeros must then be imaginary zeros.

50. Example of Descartes’ • Find the total number of zeros possible for f(x)=0 where