Chapter 6 – Polynomial Functions

1. Chapter 6 – Polynomial Functions Algebra 2

2. Warm Up

3. 6-1 Polynomial Functions • A monomial is an expression that is either a real number, a variable or a product of real numbers and variables. • A polynomial is a monomial or the sum of monomials. • The exponent of the variable in a term determines the degree of that term. • Standard form of a polynomial has the variable in descending order by degree.

4. 6-1 Polynomial Functions

5. 6-1 Polynomial Functions • The degree of a polynomial is the greatest degree of any term in the polynomial

6. 6-1 Polynomial Functions • Write each polynomial in standard form and classify it by degree.

7. 6-2 Polynomials and Linear Factors You can write a polynomial as a product of its linear factors

8. 6-2 Polynomials and Linear Factors • You can sometimes use the GCF to help factor a polynomial. The GCF will contain variables common to all terms, as well as numbers

9. 6-2 Polynomials and Linear Factors

10. 6-2 Polynomials and Linear Factors

11. 6-2 Polynomials and Linear Factors • If a linear factor of a polynomial is repeated, the zero is repeated. A repeated zero is called a multiple zero. A multiple zero has multiplicity equal to the number of times the zero occurs.

12. 6-2 Polynomials and Linear Factors

13. 6-2 Polynomials and Linear Factors • page 323 (1-11, 17-35)odd • you do NOT need to graph the functions.

14. warm up

15. 6-3 Dividing Polynomials

17. Polynomial Long Division • Two people per worksheet. • Take turns at each step, first partner decides what you multiply the divisor by, second partner agrees and does the multiplication, first partner agrees and does the subtraction, then switch for next term. • You may do the work on the worksheet, paper or the white board. If you use the white board you must have me check EACH answer as you complete it.

18. Synthetic Divison • Warm Up: • Write a polynomial function in standard form with zeros at -1, 2 and 5. • Use long division to divide: • Use long division to divide

19. Synthetic Division

24. 6-4 Solving polynomial equations

26. Solve for all three roots

27. solving using a quadratic model

30. Homework: • page 330 (227-33) odd • page 336 (13 – 31) odd,

31. warm up • Solve these equations: • 1. x3 + 125 = 0 • 2. x4 + 3x2 – 28 = 0

32. 6-5 Theorems about roots

33. 6-5 Theorems about roots

34. 6-5 Theorems about roots • To find all the roots of a polynomial: • determine the possible rational roots using the rational root theorem (ao/an) • Use synthetic division to test the possible rational roots until one divides evenly • Write the factored form and solve for all roots • Use the quadratic formula if necessary • You may need to use synthetic division more than once

35. 6-5 Theorems about roots

36. 6-5 Theorems about roots • Warm Up • Find the polynomial equation in standard form that has roots at -5, -4 and 3 • Find f(-2) for f(x) = x4 – 2x3 +4x2 + x + 1 using synthetic division • Solve x4 – 100 = 0

37. 6-5 Theorems about roots • Practice Problem: • List all the possible rational roots of • 3x3 + x2 – 15x – 5 = 0 Use synthetic division to determine which of these is a root Factor and solve for the rest of the roots of the equation.

38. 6-5 Theorems about roots

39. 6-5 Theorems about roots • A third degree polynomial has roots 2 and √3.Write the polynomial in standard form.

40. 6-5 Theorems about roots

41. 6-5 Theorems about roots

42. 6-5 Theorems about roots

43. 6-5 Theorems about roots

44. 6-5 Theorems about roots • Homework • p 345 (11-23) odd

45. 6-6 Fundamental Theorem of Algebra

46. 6-6 Fundamental Theorem of Algebra

47. 6-6 Fundamental Theorem of Algebra

48. 6-7 Permutations and Combinations

50. A selection of items in which order does not matter is called a combination