1. Chapter 2 notes from powerpoints

2. Synthetic division – and basic definitions
Sections 1 and 2

3. No! No! Definition of a Polynomial Function:
Let n be a nonnegative integer and let an, an-1, …, a2, a1, a0 be real numbers.
The following function is called a polynomial function of x with degree n.
No!
No!
In polynomial functions, THE EXPONENTS ON THE VARIABLE CANNOT BE FRACTIONS AND CANNOT BE NEGATIVE.

4. 2.1 Polynomials
Constant Term
Linear Term
Quadratic Term
n-1 Term

5. Name of Polynomial Function
Classifying Polynomials
Polynomials are often classified by their degree. The degree of a polynomial is the highest degree of its terms.
Degree
Name of Polynomial Function
Example
Zero
f(x) = -3
First
f(x) = 2x + 5
Second
f(x) = 3x2 –5x + 2
Third
f(x) = x3 – 2x –1
Fourth
f(x) = x4 –3x3 +7x-6
Fifth
f(x) = 2x5 + 3x4 – x3+x2
Constant
Linear
Quadratic
Cubic
Quartic
Quintic

6. Any value of x for which 𝑃 𝑥 =0 is a root of the equation and a zero of the function.
0=𝑥 2 −4=(𝑥+2)(𝑥−2)
𝑃 𝑥 = 𝑥 2 −4
±2 are the zeros of the function

7. Main Ideas:
Synthetic Division

8. Synthetic Division Main Ideas 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑥−2 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑃(𝑥)
Is 𝑥−2 a factor?
The factor theorem states that if a is a constant, then 𝑥−𝑎 is a factor of polynomial 𝑃 𝑥 if and only if 𝑃 𝑎 =0
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑥−2
𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑃(𝑥)

9. Synthetic Division Main Ideas Is 𝑥+2 a factor? 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑥+2
𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑆(𝑥)
The factor theorem states that if a is a constant, then 𝑥−𝑎 is a factor of polynomial 𝑃 𝑥 if and only if 𝑃 𝑎 =0

10. Section 2.2 Synthetic Division; The Remainder and Factor Theorems
Objective: To use synthetic division and to apply the remainder and factor theorems.

11. Vocabulary
4 is not a factor of 23, because if when 23 is divided by 4 the remainder is not zero.
When 23 is divided by 4, the quotient is 5 and the remainder is 3.

12. The Remainder Theorem
When a polynomial P(x) is divided by x – a, the remainder is P(a)
Remainder =P(a)

13. The Factor Theorem ⇔
if and only if

14. SECTion 3- graphing polynomials

15. Starts Up,
Ends Up
Starts Down,
Ends Down

16. Starts Down,
Ends Up
Starts Up,
Ends Down

17. 2.3 Graphing Polynomial Functions

18. 2.3 Graphing Polynomial Functions
Zeros: -1, 1, 2

19. 2.3 Graphing Polynomial Functions

20. 2.3 Graphing Polynomial Functions

21. 2.3 Graphing Polynomial Functions

22. 2.3 Graphing Polynomial Functions

23. 2.3 Graphing Polynomial Functions

24. 2.3 Graphing Polynomial Functions

25. Effect of A Squared Factor

26. Effect of A Squared Factor

27. Homework
Section 2.2
Page 61 #1-25 odds

28. Effect of A Squared Factor

29. Effect of A Squared Factor

30. Effect of A Cubed Factor

31. Effect of A Cubed Factor

32. Effect of A Cubed Factor

33. Effect of A Cubed Factor

34. Graphing review
For test review: 9/26

35. examples Determine end behavior and orientation
I can graph polynomials
examples
Determine end behavior and orientation
𝐟 𝐱 = 𝟔𝒙 𝟕 −𝟑 𝒙 𝟑 + 𝒙 𝟐 −𝟏𝟐
𝐟 𝐱 =−𝟑 𝒙 𝟔 −𝒙+𝟏
Find the x- and y-intercepts
𝐲=𝟒 𝒙−𝟓 𝟐 (𝒙+𝟏)(𝟐𝒙−𝟓)

36. Warm Up 𝑥→−∞, 𝑓 𝑥 →−∞ 𝑥→∞, 𝑓 𝑥 →∞ 𝑥→−∞, 𝑓 𝑥 →−∞ 𝑥→∞, 𝑓 𝑥 →−∞
Determine end behavior and orientation
𝒇 𝒙 =𝟔𝒙 𝟕 −𝟑 𝒙 𝟑 + 𝒙 𝟐 −𝟏𝟐
𝐟 𝐱 =−𝟑 𝒙 𝟔 −𝒙+𝟏
𝑥→−∞, 𝑓 𝑥 →−∞
𝑥→∞, 𝑓 𝑥 →∞
𝑥→−∞, 𝑓 𝑥 →−∞
𝑥→∞, 𝑓 𝑥 →−∞

37. Use Zero Product Property!
Warm Up
Find the x- and y-intercepts
y=4 𝑥−5 2 (𝑥+1)(2𝑥−5)
X-intercepts
Use Zero Product Property!
𝑥−5=0 𝑥+1= 𝑥−5=0
𝑥= 𝑥=− 𝑥=5/2
y-intercept
Plug in 0 for x
y=4 0− −5
y=4 −5 2 (1)(−5)
𝑦=−500

38. Graphing Polynomials
𝑦= 𝑥−1 2 (𝑥+1) 𝒙 𝒚 −2 −1 1 2

39. Graphing Polynomials
𝑦= (𝑥−1) 2 (𝑥+1) 2 𝒙 𝒚 −2 −1 1 2

40. Graphing a Polynomial Steps Find the roots (x-intercepts)
Example:
𝒚=𝟐 𝒙+𝟏 𝒙−𝟐 (𝒙+𝟖)
Find the roots
(x-intercepts)
2) Find the y-intercept
𝑥+1=0 𝑥−2=0 𝑥+8=0 𝑥=−1 𝑥=2 𝑥=−8 (−1,0) (2,0) (−8,0) 𝑦=2(0+1)(0−2)(0+8) 𝑦=2 1 −2 8 𝑦=−32 (0,−32)

41. Graphing a Polynomial As 𝑥→−∞, 𝑓 𝑥 →−∞ As 𝑥→∞, 𝑓 𝑥 →∞ Steps
Example:
𝒚=𝟐 𝒙+𝟏 𝒙−𝟐 (𝒙+𝟖)
3) Determine orientation and end behavior
Odd degree, positive leading term
As 𝑥→−∞, 𝑓 𝑥 →−∞
As 𝑥→∞, 𝑓 𝑥 →∞

42. Graphing a Polynomial As 𝑥→−∞, 𝑓 𝑥 →−∞ As 𝑥→∞, 𝑓 𝑥 →∞ Steps 4) Graph
Example:
𝒚=𝟐 𝒙+𝟏 𝒙−𝟐 (𝒙+𝟖)
4) Graph
−1,0
(2,0)
(−8,0)
(0,−32)
As 𝑥→−∞,
𝑓 𝑥 →−∞
As 𝑥→∞,
𝑓 𝑥 →∞

43. Graphing a Polynomial Steps Find the roots (x-intercepts)
Example 2:
𝒚= −𝒙+𝟓 (𝒙−𝟏) 𝟐 𝒙+𝟐
Find the roots
(x-intercepts)
2) Find the y-intercept
−𝑥+5=0 𝑥−1=0 𝑥+2=0 𝑥=5 𝑥=1 𝑥=−2 (5,0) (1,0) (−2,0) 𝑦= −0+5 (0−1) 𝑦= 5 (−1) 2 2 𝑦=10 (0,10)

44. Even degree, negative leading term
Graphing a Polynomial
Steps
Example 2:
𝒚= −𝒙+𝟓 (𝒙−𝟏) 𝟐 𝒙+𝟐
3) Determine orientation and end behavior
Even degree, negative leading term
As 𝑥→−∞, 𝑓 𝑥 →−∞
As 𝑥→∞, 𝑓 𝑥 →−∞

45. Graphing a Polynomial As 𝑥→−∞, 𝑓 𝑥 →−∞ As 𝑥→∞, Steps 4) Graph 5,0
Example:
𝒚= −𝒙+𝟓 (𝒙−𝟏) 𝟐 𝒙+𝟐
4) Graph
5,0
(1,0)
(−2,0)
(0,10)
As 𝑥→−∞,
𝑓 𝑥 →−∞
As 𝑥→∞,

46. Try on your own: Graphing a Polynomial
Steps
Example 3:
𝒚=−𝒙 𝒙+𝟏 (𝒙−𝟑)
Find the roots
(x-intercepts)
2) Find the y-intercept
−𝑥=0 𝑥+1=0 𝑥−3=0 𝑥=0 𝑥=−1 𝑥=3 (0,0) (−1,0) (3,0) 𝑦= −3 𝑦= 0 (1) −3 𝑦=0 (0,0)

47. Graphing a Polynomial As 𝑥→−∞, 𝑓 𝑥 →∞ As 𝑥→∞, 𝑓 𝑥 →−∞ Steps
Example:
𝒚=−𝒙 𝒙+𝟏 (𝒙−𝟑)
3) Determine orientation and end behavior
Odd degree, negative leading term
As 𝑥→−∞, 𝑓 𝑥 →∞
As 𝑥→∞, 𝑓 𝑥 →−∞

48. Warm Up Graph: 𝒚=−𝒙 𝒙+𝟏 (𝒙−𝟑) x – intercepts: y – intercept:
I can graph a polynomial
Warm Up
Graph: 𝒚=−𝒙 𝒙+𝟏 (𝒙−𝟑)
x – intercepts:
y – intercept:
End Behavior:

49. Try on your own: Graphing a Polynomial
Steps
Example 3:
𝒚=−𝒙 𝒙+𝟏 (𝒙−𝟑)
Find the roots
(x-intercepts)
2) Find the y-intercept
−𝑥=0 𝑥+1=0 𝑥−3=0 𝑥=0 𝑥=−1 𝑥=3 (0,0) (−1,0) (3,0) 𝑦= −3 𝑦= 0 (1) −3 𝑦=0 (0,0)

50. Graphing a Polynomial As 𝑥→−∞, 𝑓 𝑥 →∞ As 𝑥→∞, 𝑓 𝑥 →−∞ Steps
Example:
𝒚=−𝒙 𝒙+𝟏 (𝒙−𝟑)
3) Determine orientation and end behavior
Odd degree, negative leading term
As 𝑥→−∞, 𝑓 𝑥 →∞
As 𝑥→∞, 𝑓 𝑥 →−∞

51. X – Intercepts: (0,0) (−1,0) ,0
Y – Intercept:(0,0)
End Behavior: Odd degree, negative leading term
As 𝑥→−∞, 𝑓 𝑥 →∞
As 𝑥→∞, 𝑓 𝑥 →−∞

52. 2.6 Solving Polynomial Functions by Factoring

53. Grouping best to use if:
P(x) is a cubic.
P(x) has 4 terms.
Pair up terms.
Factor within the terms
Factor again.

54. Try it: 𝑥 3 +6 𝑥 2 −4𝑥−24=0
𝑥=−6, ±2

55. 𝑥 3 +6 𝑥 2 −4𝑥−24=0 𝑥 3 +6 𝑥 2 + −4𝑥−24 =0 𝑥 2 𝑥+6 −4 𝑥+6 =0
𝑥 3 +6 𝑥 2 + −4𝑥−24 =0
𝑥 2 𝑥+6 −4 𝑥+6 =0
𝑥+6 𝑥 2 −4 =0
𝑥+6 𝑥+2 𝑥−2 =0
𝑥+6=0, 𝑥−2=0, 𝑥+2=0
𝑥=−6, 𝑥=2, 𝑥=−2

57. 2 𝑥 4 − 𝑥 2 −3=0
Let 𝑢= 𝑥 2
Then: 2 𝑥 4 − 𝑥 2 −3=0, becomes:
2 𝑢 2 −𝑢−3=0
2𝑢−3 𝑢+1 =0
𝑢= 3 2 , 𝑢=−1
Remember 𝑢= 𝑥 2 , so solving for x:
3 2 = 𝑥 2 , 𝑥=± = = =± , and
−1= 𝑥 2 𝑠𝑜 𝑥=±𝑖

58. When to use quadratic form
The polynomial has 3 terms
The degree is an even number (though this is not required)
The exponent of the middle term is half the degree (necessary)
The 3rd term is constant
Examples: 𝑥 8 + 𝑥 4 +20, 𝑥 22 + 𝑥 , 𝑥 13 + 𝑥 6.5 −215

59. Try it: 2𝑥 4 =−7 𝑥 2 +15
𝑥=± , ±𝑖 5

60. Classwork
Class Exercises
Page 60 #1-5

61. Homework
Page 61 #5,7,13,19,23 Page 66 #3,9,11,13,15,21,23,27,29

62. Homework
Page 83 #1, 3, 9, 10, and 13 – 34 every 3rd

63. Classwork
Page 83 #1-6