1. Polynomials
Chapter 6

2. 6.1 - Polynomial Functions

3. Objectives By the end of today, you will be able to…
Classify polynomials
Model data using polynomial functions

5. Vocabulary A polynomial is a monomial or the sum of monomials.
The exponent of the variable in a term determines the degree of that polynomial.
Ordering the terms by descending order by degree. This order demonstrates the standard form of a polynomial.
P(x) = 2x³ - 5x² - 2x + 5
Leading Coefficient
Constant Term
Cubic Term
Quadratic Term
Linear Term

6. Standard Form of a Polynomial
For example: P(x) = 2x3 – 5x2 – 2x + 5
Polynomial
Standard Form Polynomial

7. Parts of a Polynomial P(x) = 2x3 – 5x2 – 2x + 5 Leading Coefficient:
Cubic Term:
Quadratic Term:
Linear Term:
Constant Term:

8. Parts of a Polynomial P(x) = 4x2 + 9x3 + 5 – 3x Leading Coefficient:
Cubic Term:
Quadratic Term:
Linear Term:
Constant Tem:

9. Classifying Polynomials
We can classify polynomials in two ways:
By the number of terms
# of Terms
Name
Example 1
Monomial 3x 2
Binomials
2x2 + 5 3
Trinomial
2x3 + 3x + 4 4
Polynomial with 4 terms
2x3 – 4x2 + 5x + 4

10. Classifying Polynomials
2) By the degree of the polynomial (or the largest degree of any term of the polynomial.
Degree
Name
Example
Constant 7 1
Linear
2x + 5 2
Quadratic
2x2 3
Cubic
2x3 – 4x2 + 5x + 4 4
Quartic
x4 + 3x2 5
Quintic
3x5 – 3x + 7

11. Classifying Polynomials
Write each polynomial in standard form. Then classify it by degree AND number of terms.
-7x2 + 8x x2 + 4x + 4x3 + 4
3. 4x + 3x + x – 3x

12. Cubic Regression STAT  Edit x-values in L1, y-values in L2 STAT CALC
We have already discussed regression for linear functions, and quadratic functions. We can also determine the Cubic model for a given set of points using Cubic Regression.
STAT  Edit
x-values in L1, y-values in L2
STAT CALC
6:CubicReg

13. Cubic Regression Find the cubic model for each function:
(-1,3), (0,0), (1,-1), (2,0)
(10, 0), (11,121), (12, 288), (13,507)

14. Picking a Model
Given Data, we need to decide which type of model is the best fit.

15. Comparing Models
Using a graphing calculator, determine whether a linear, quadratic, or cubic model best fits the values in the table.
x y
0 2.8
2 5
4 6
6 5.5
8 4
Enter the data. Use the LinReg, QuadReg, and CubicReg
options of a graphing calculator to find the best-fitting model
for each polynomial classification.
Graph each model and compare.
Linear model
Quadratic model
Cubic model
The quadratic model appears to best fit the given values.

16. Polynomial
Models
You have already used lines and parabolas to model data. Sometimes you can fit data more closely by using a polynomial model of degree three or greater.
Using a graphing calculator, determine whether a linear model, a quadratic model, or a cubic model best fits the values in the table. x 5 10 15 20 y
10.1
2.8
8.1
16.0
17.8

17. 6.2 - Polynomials & Linear Factors

18. Factored Form
The Factored form of a polynomial is a polynomial broken down into all linear factors.
We can use the distributive property to go from factor form to standard form.

19. Factored to Standard (x+1)(x+2)(x+3)
Write the following polynomial in standard form:
(x+1)(x+2)(x+3)

20. Factored to Standard (x+1)(x+1)(x+2)
Write the following polynomial in standard form:
(x+1)(x+1)(x+2)

21. Factored to Standard x(x+5)2
Write the following polynomial in standard form:
x(x+5)2

22. Standard to Factored form
Factor out the GCF of all the terms
Factor the Quadratic
Example: 2x3 + 10x2 + 12x

23. Standard to Factored form
Write the following in Factored Form 3x3 – 3x2 – 36x

24. Standard to Factored form
Write the following in Factored Form x3 – 36x

25. The Graph of a Cubic

26. Relative Maximum: The greatest Y-value of the points in a region.
Relative Minimum: The least Y-value of the points in a region.
Zeros: Place where the graph crosses x-axis
y-intercept: Place where the graph crosses y-axis
Vocabulary

27. Relative Max and Min f(x) = x3 +4x2 – 5x f(x) = -x3 – 7x2 – 18x
Find the relative max and min of the following polynomials:
f(x) = x3 +4x2 – 5x
Relative min:
Relative max:
f(x) = -x3 – 7x2 – 18x
(-3.2 , 24.2) (.5, -1.4)
(7.1, 5) ( 3.6, -16.9)
Calculator:
2nd  CALC  Min or Max
Use a left bound and a right bound for each min or max.

28. Finding Zeros
When a Polynomial is in factored form, it is easy to find the zeros, or where the graph crosses the x-axis. EX: Find the Zeros of y = (x+4)(x – 3)

29. Factor Theorem
The Expression x – a is a linear factor of a polynomial if and only if the value a is a zero of the related polynomial function.

30. Find the Zeros Find the Zeros of the Polynomial Function.
y = (x – 2)(x + 1)(x + 3)
y = (x – 7)(x – 5)(x – 3)

31. Writing a Polynomial Function
Give the zeros -2, 3, and -1, write a polynomial function. Then classify it by degree and number of terms.
Give the zeros 5, -1, and -2, write a polynomial function. Then classify it by degree and number of terms.

32. Repeated Zeros
A repeated zero is called a MULITIPLE ZERO. A multiple zero has a MULTIPLICITY equal to the number of times the zero repeats.

33. Find the Multiplicity of a Zero
Find any multiple zeros and their multiplicity
y = x4 + 6x3 + 8x2

34. Find the Multiplicity of a Zero
Find any multiple zeros and their multiplicity
y = (x – 2)(x + 1)(x + 1)2
y = x3 – 4x2 + 4x

35. 6.3 Dividing Polynomials

36. Vocabulary Dividend: number being divided
Divisor: number you are dividing by
Quotient: number you get when you divide
Remainder: the number left over if it does not divide evenly
Factors: the DIVISOR and QUOTIENT are FACTORS if there is no remainder

37. Long Division
Divide WITHOUT a calculator!! 1. 2.

38. Steps for Dividing

39. Using Long Division on Polynomials
Using the same steps, divide.

40. Using Long Division on Polynomials
Using the same steps, divide.

41. Using Long Division on Polynomials
Using the same steps, divide.

42. Synthetic Division

43. Synthetic Division
Step 1: Switch the sign of the constant term in the divisor. Write the coefficients of the polynomial in standard form. Step 2: Bring down the first coefficient. Step 3: Multiply the first coefficient by the new divisor. Step 4: Repeat step 3 until remainder is found.

44. Example
Use Synthetic division to divide 3x3 – 4x2 + 2x – 1 by x + 1

45. Example
Use Synthetic division to divide X3 + 4x2 + x – 6 by x + 1

46. Example
Use Synthetic division to divide X3 + 3x2 – x – 3 by x – 1

47. Remainder Theorem
Remainder Theorem: If a polynomial P(x) is divided by (x – a), where a is a constant, then the remainder is P(a).

48. Using the Remainder Theorem
Find P(-4) for P(x) = x4 – 5x2 + 4x + 12.

49. 6.4 Solving Polynomials by Graphing

50. Solving by Graphing: 1st Way
Solutions are zeros on a graph
Step 1: Solve for zero on one side of the equation.
Step 2: Graph the equation
Step 3: Find the Zeros using 2nd  CALC
(Find each zero individually)

51. Solving by Graphing: 2nd Way
Step 1: Graph both sides of the equal sign as two separate equations in y1 and y2.
Use 2nd  CALC  Intersect to find the x values at the points of intersection

52. Solve by Graphing
x3 + 3x2 = x + 3
x3 – 4x2 – 7x = -10

53. Solve by Graphing
x3 + 6x2 + 11x + 6 = 0

54. Solving by Factoring

55. Factoring Sum and Difference
Factoring cubic equations: Note: The second factor is prime (cannot be factored anymore)

56. Factor:
x3 - 8
27x3 + 1

57. You Try! Factor:
x3 + 64
8x3 - 1
8x3 - 27

58. Solving a Polynomial Equation

59. Solving By Factoring
Remember: Once a polynomial is in factored form, we can set each factor equal to zero and solve. 4x3 – 8x2 + 4x = 0

60. Solve by factoring:
1. 2x3 + 5x2 = 7x 2. x2 – 8x + 7 = 0

61. Using the patterns to Solve
So solve cubic sum and differences use our pattern to factor then solve. X3 – 8 = 0

62. Using the patterns to Solve
x3 – 64 = 0

63. Using the patterns to Solve
x = 0

64. Factoring by Using Quadratic Form

65. Factoring by using Quadratic Form
x4 – 2x2 – 8

66. Factoring by using Quadratic Form
x4 + 7x2 + 6

67. Factoring by using Quadratic Form
x4 – 3x2 – 10

68. Solving Using Quadratic Form
x4 – x2 = 12

69. 6.5 Theorems About Roots

70. The Degree
Remember: the degree of a polynomial is the highest exponent. The Degree also tells us the number of Solutions (Including Real AND Imaginary)

71. Solutions/Roots
How many solutions will each equation have? What are they?
x3 – 6x2 – 16x = 0
x = 0

72. Solving by Graphing
Solving by Graphing ONLY works for REAL SOLUTIONS. You cannot find Imaginary solutions from a Graph. Roots: This is another word for zeros or solutions.

73. If p/q is a rational root (solution) then:
Rational Root Theorem
If p/q is a rational root (solution) then:
p must be a factor of the constant
and
q must be a factor of the leading coefficient

74. Example
x3 – 5x2 - 2x + 24 = 0
Lets look at the graph to find the solutions
Factored  (x + 2)(x – 3)(x – 4) = 0
Note: Roots are all factors of 24 (the constant term) since a = 1.

75. Example
24x3 – 22x2 - 5x + 6 = 0
Lets look at the graph to find the solutions:
Factored  (x + ½ )(x – ⅔)(x – ¾ ) = 0
1,2, and 3 (the numerators) are all factors of 6 (the constant).
2, 3, and 4 (the denominators) are all factors of 24 (the leading coefficient).

76. 8) x3 – 5x2 + 7x – 35 = 0

77. 10) 4x3 + 16x2 -22x -10 = 0

78. Irrational Root Theorem Square Root Solutions come in PAIRS: If x2 = c then x = ± √c If √ is a solution so is -√ Imaginary Root Theorem If a + bi is a solution, so is a – bi

79. Recall
Solve the following by taking the square root: X2 – 49 = 0 X = 0

80. Using the Theorems √5 2. -√6 3. 2 – i 4. 2 - √3
Given one Root, find the other root!
√ √6
3. 2 – i √3

81. Zeros to Factors
If a is a zero, then (x – a) is a factor!! When you have factors (x – a)(x – b) = x2 + (a+b)x + (ab) SUM PRODUCT

82. Examples Find a 2nd degree equation with roots 2 and 3
(x - _______)(x - ______)
2. Find a 2nd degree equation with roots -1 and 6

83. Example
Find a 2nd degree equation with roots ±√7

84. Examples Find a 2nd degree equation with roots ±2√5
Find a 2nd degree equation with roots ±6i

85. Examples
Find a 2nd degree equation with a root of 7 + i

86. Example
Find a 3rd degree equation with roots 4 and 3i (x - _______)(x - ______)(x - ______)

87. Example
Find a third degree polynomial equation with roots 3 and 1 + i.