C. Higher Functions

1. C. Higher Functions Pre-Calculus 30

2. PC30.10 • Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

3. Key Terms

5. 1. Characteristics of Polynomial Functions • PC30.10 • Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

6. 1. Characteristics of Polynomial Functions

7. What is a Polynomial Function?

8. A polynomial function has the form • Where • n = is a whole number • x = is a variable • The coefficients an to a0 are real numbers • The degree of the poly function is n, the exponent of the greatest power of x • The leading coefficient is an the coefficient of the greatest power of x • The constant term is a0

9. Example 1

10. Types of Polynomial Functions:

12. Each graph has at least one less change of direction then the degree of its function.

13. Characteristics of Polynomial Functions:

16. Example 2

17. Example 3

18. Key Ideas p.113

21. Practice • Ex. 3.1 (p.114) #1-4 odds in each, 5-10 # 1-4 odds in each, 5-13 odds

22. 2. Finding X-Intercepts • PC30.10 • Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

23. 2. Finding X-Intercepts • When only given the equation of a function what are some strategies that we have use to find x-intercepts? • Sub in a zero and solve • Factor • Quadratic Formula • Decompostion

24. Which of these strategies will work if our polynomial function have a degree greater than 2?

25. In this section we will look at some methods to completely factor a polynomial function with a degree greater than 2 in order to find the zeros (x-intercepts).

26. Long Division • Long division of a polynomial is done just like your do with numbers but now you have variables

27. You use long division to divide a polynomial by a binomial: • The dividend, P(x), which is the polynomial that is being divided • The divisor, x-a, which is the binomial being divided into the polynomial • The quotient, Q(x), which is the expression that results from the division • The remainder, R, which is what is left over when the division is done.

28. To Check the division of the polynomial, verify the statement

29. Example 1

30. Example 2

31. If we graphed the polynomial function from example 2 what would the x-intercepts be?

32. Long division gives is factors of the polynomial function which when set equal to zero and solved are x-intercepts, zeros, or roots.

33. Synthetic Division: • A short form of division that uses only the coefficients of the terms • It involves fewer calculations

34. Example 3

35. Remainder Theorem: • When a polynomial P(x) is divided by a binomial x-a, the remainder is P(a) • If the remainder is 0 then the binomial x-a is a factor of P(x) • If the remainder is not 0 then the binomial x-a is NOT a factor of P(x)

36. Example 4

37. Key Ideas p.123

38. Practice • Ex. 3.2 (p.124) #1,2,3-7 odds in each, 8-13 #2, 3-7 odds in each, 8, 9-17 odds

39. 3. Factoring Completely • PC30.10 • Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

40. 3. Factoring Completely • Last day we looked at how dividing polynomial functions by a binomial shows us if that binomial is a factor or not • We also discussed how we want our equations in factored form because that gives is the zeros/roots/x-intercepts • Today we are going to extend that idea

41. Factor Theorem: • The factor theorem states that x-a is a factor of a polynomial P(x) if and only if P(a)=0 • If and only if means that the result works both ways. That is, • If x-a is a factor then P(a)=0 • If P(a)=0, then x-a is a factor of a polynomial P(x)

42. For example, which binomial are factors of the polynomial ? • x-1 • x-2 • x+2 • x+3

43. The zeros of a polynomial function are directly related to the x-intercepts • If we graph we will see the zero/x-intercepts are at x=1, x= -2, and x= -3 • The corresponding factors are x-1, x+2, x+3 • So to find the x-intercepts given the polynomial functions equation we have to get the equation in fully factored form.

44. Example 1

45. When factoring a polynomial function sometimes the most difficult part is deciding which values of “a” we should use when using long division, synthetic division or factor theorem.

46. Consider the Polynomial function • We want P(x)=0 when x=a so……..

47. This is referred to as the Integral Zero Theorem • The integral zero theorem describes the relationship between the factors and the constant term of a polynomial. • The theorem states that if x-a is a factor of a polynomial P(x) with integral coefficients, then “a” is a factor of the constant term of P(x) and x=a is a integral zero of P(x).

48. Example 2

49. Factor by Grouping: • If a polynomial P(x) has an even number of terms, it may be possible to group tow terms at a time and remove a common factor • If the binomial that results from common factoring is the same for each pair of terms, then P(x) may be factored by grouping • Will not always work!!!!

50. Steps to factoring Polynomial Functions: 1. Use the integral zero theorem to list possible integer values for zeros 2. You can use the factor theorem to determine if the values that are zeros (this take a lot of time so I don’t suggest it) 2. Use one type of division to determine all the factors 3. Write equation in factored form