1. Ms. C. Taylor Common Core Math 3

2. Warm-Up

3. Polynomials

4. Handout  Fill in the handout on Investigation of Polynomial Functions

5. Relative Maxima or Minima

6. Higher Degree Polynomial Functions and Graphs  a n is called the leading coefficient  n is the degree of the polynomial  a 0 is called the constant term Polynomial Function A polynomial function of degree n in the variable x is a function defined by where each a i is real, a n  0, and n is a whole number.

7. Polynomial Functions f(x) = 3 ConstantFunction Degree = 0 Maximum Number of Zeros: 0

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. The Leading Coefficient Test As x increases or decreases without bound, the graph of the polynomial function f (x)  a n x n  a n-1 x n-1  a n-2 x n-2  …  a 1 x  a 0 (a n  0) eventually rises or falls. In particular, For n odd:a n  0 a n  0 As x increases or decreases without bound, the graph of the polynomial function f (x)  a n x n  a n-1 x n-1  a n-2 x n-2  …  a 1 x  a 0 (a n  0) eventually rises or falls. In particular, For n odd:a n  0 a n  0 If the leading coefficient is positive, the graph falls to the left and rises to the right. If the leading coefficient is negative, the graph rises to the left and falls to the right. Rises right Falls left Falls right Rises left

13. As x increases or decreases without bound, the graph of the polynomial function f (x)  a n x n  a n-1 x n-1  a n-2 x n-2  …  a 1 x  a 0 (a n  0) eventually rises or falls. In particular, For n even: a n  0 a n  0 As x increases or decreases without bound, the graph of the polynomial function f (x)  a n x n  a n-1 x n-1  a n-2 x n-2  …  a 1 x  a 0 (a n  0) eventually rises or falls. In particular, For n even: a n  0 a n  0 If the leading coefficient is positive, the graph rises to the left and to the right. If the leading coefficient is negative, the graph falls to the left and to the right. Rises right Rises left Falls left Falls right The Leading Coefficient Test

14. Example Use the Leading Coefficient Test to determine the end behavior of the graph of f (x)  x 3  3x 2  x  3. Falls left y Rises right x

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

16. x-Intercepts (Real Zeros ) Number Of x-Intercepts of a Polynomial Function A polynomial function of degree n will have a maximum of n x- intercepts (real zeros). Find all zeros of f (x) = -x 4 + 4x 3 - 4x 2.  x 4  4x 3  4x 2  We now have a polynomial equation. x 4  4x 3  4x 2  0 Multiply both sides by  1. (optional step) x 2 (x 2  4x  4)  0 Factor out x 2. x 2 (x  2) 2  0 Factor completely. x 2  or (x  2) 2  0 Set each factor equal to zero. x  0 x  2 Solve for x. (0,0) (2,0)

17. Multiplicity and x-Intercepts If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.

18. Example  Find the x-intercepts and multiplicity of f(x) =2(x+2) 2 (x-3 )  Zeros are at (-2,0) (3,0)

19. Extrema  Turning points – where the graph of a function changes from increasing to decreasing or vice versa. The number of turning points of the graph of a polynomial function of degree n  1 is at most n – 1.  Local maximum point – highest point or “peak” in an interval function values at these points are called local maxima  Local minimum point – lowest point or “valley” in an interval function values at these points are called local minima  Extrema – plural of extremum, includes all local maxima and local minima

20. Extrema

21. Warm-Up

22. End Behavior DegreeLeading CoefficientEnd Behavior EvenPositive EvenNegative OddPositive OddNegative

23. Domain & Range

24.  The “h” will shift the graph left or right depending on sign  The “k” will shift the graph up or down depending on the sign.

25. Remainder Theorem  If I divide the polynomial by the factor given then I will have a remainder other than zero.  If you plug in the constant of the factor into the equation then you should come up with the remainder when doing synthetic division.

26. Examples-Remainder Theorem

27. Warm-Up

28. Factor Theorem  In order to prove that (x-a) is a factor of the polynomial f(a)=0.  In other words, the remainder from synthetic division has to be zero.

29. Examples

30. Fundamental Theorem of Algebra  Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.

31. Rational Zero Theorem

32. Examples

33. Warm-Up

34. Rational Equations  Step 1: Set all denominators equal to zero and solve.  Step 2: Find the CD (Factor Denom.)  Step 3: Look for “what is missing” & multiply each fraction by the appropriate expression. DUMP the denominators.  Step 4: Solve for x.  Step 5: Check to be sure that x doesn’t equal any number from step 1.

35. Rational Equations

36. Warm-Up

37. Literal Equations

38. Warm-Up

39. Practice  An open box is to be made from a rectangular piece of material 20 inches by 12 inches by cutting equal squares from the corners and turning up the sides.  What is the maximum volume that the box can hold?

40. Practice  An open box is to be made from a rectangular piece of material 15 inches by 10 inches by cutting equal squares from the corners and turning up the sides.  What is the maximum volume that the box can hold?

41. Practice  An open box is to be made from a rectangular piece of material 6 inches by 1 inch by cutting equal squares from the corners and turning up the sides.  What is the maximum volume that the box can hold?

42. Practice  An open box is to be made from a rectangular piece of material 25 inches by 14 inches by cutting equal squares from the corners and turning up the sides.  What is the maximum volume that the box can hold?

43. Practice  An open box is to be made from a rectangular piece of material 35 inches by 22 inches by cutting equal squares from the corners and turning up the sides.  What is the maximum volume that the box can hold?

44. Practice  An open box is to be made from a rectangular piece of material 40 inches by 24 inches by cutting equal squares from the corners and turning up the sides.  What is the maximum volume that the box can hold?

45. Warm-Up

46. Finding k in a Polynomial  Substitute the value from the factor in for x and then set the equation equal to zero.  Check by using synthetic division to make sure that you get a remainder of 0.

47. Examples

48. Practice

49. Warm-Up

50. To write the polynomial equation given the zeros:

51.  Step 2: Multiply the factors that have i’s or first  Remember: Imaginary numbers and irrational numbers come in conjugate pairs (ex: 2+3i, 2-3i)  Step 3: Multiply the trinomial and binomial by the box

52. When you are given the zeros/roots: write them as factors: (x - )(x - )(x - ) x = {,, }2 -2 Change the signs if you can: (x + 2)(x +1)(x-2)

53. Now write your equation by multiplying out! (x + 2)(x +1)(x-2) Box! Box! AGAIN!

54. 2. When you are given the zeros/roots: write them as factors: (x - )( x - ) x = {, } 3/4 -2 Change the signs if you can: (x + 2)(4x-3) 4x - 3

55. Now write your equation by multiplying out! (x + 2)(4x -3) Box!

56. 3. When you are given the zeros/roots: write them as factors: (x - )(x - )(x - ) x = {,, }3 -2 5 Change the signs if you have to: (x -5)(x +2)(x-3)

57. Now write your equation by multiplying out! (x -5)(x +2)(x-3) Box! Box! AGAIN!

58. Warm-Up