1. S.A. = 2x(x-1)(x+1) + 2x(x-1) + 2x(x+1)
Warm-up 1/28/08
Find the surface area and volume of a box with one edge of unknown length, a second edge one unit longer, and a third edge one unit shorter. Write your answers as a polynomial.
Surface Area = 2B + Ph
S.A. = 2x(x-1)(x+1) + 2x(x-1) + 2x(x+1)
S.A. = 6x2 – 2
Volume = lwh
V = x(x-1)(x+1)
V = x3 - x
12a8
4a3
-18m3n3 X7
1/p4q2
s7t4

2. Warm-up 1/29/08 Graph f(x) = x4 – 2x2 – 1
At what value(s) of x does the function f attain its minimum value?
What is the minimum value?

3. Topic:
Polynomials and Polynomial Functions
Key Learning(s):
Graph and interpret graphs of quadratics and higher degree polynomial functions
Unit Essential Question (UEQ):
How do you construct and interpret polynomials that model real situations?

4. Concept I:
Polynomial Models
Lesson Essential Question (LEQ):
How do you create and interpret polynomial models?
Vocabulary:
Polynomial Polynomial function
Degree Monomial
Leading coefficients Binomial
Standard form Trinomial

5. Concept II
Graphs of Polynomial Functions
Lesson Essential Question (LEQ):
How do you find important points on a polynomial function?
Vocabulary:
Maximum value Relative min.
Minimum value Zeros (roots)
Extreme values (extrema)
Relative extremum End Behavior
Relative maximum

6. Concept III
Finding Polynomial Models
Lesson Essential Question (LEQ):
How do you find a polynomial model for a given set of data?
Vocabulary:
Tetrahedral number
Polynomial difference theorem

7. Concept IV
Useful Theorems
Lesson Essential Question (LEQ):
How are the remainder theorem and factor theorem used to solve polynomial functions?
How is the fundamental theorem of algebra a useful tool for solving polynomials?
Vocabulary:
Remainder theorem multiplicity of a
Factor theorem zero
Fundamental theorem of algebra

8. Concept V
Factoring Polynomials
Lesson Essential Question (LEQ):
How do you solve polynomial equations by factoring?
Vocabulary:
Sums and differences of cubes

9. §9.1: Polynomial Models
How do you create and interpret Polynomial models?
Polynomial:
Expression in the form:
anxn + an-1xn-1 + an-2xn-2 + a1x + a0
Ex) a4x4 + a3x3 + a2x2 + a1x + a0

10. Polynomial Function –
Any sum or difference of power functions and constants.
Degree –
The exponent determines the degree of the function.
Standard form –
All like terms are combined.
The function is written in descending order by degree.
Leading Coefficient

11. Functions like y = x4 and w = .084C3 are power functions.
A function with the form y = axn, where a≠0 and n is a positive integer.
Even functions –
Have the y-axis as the axis of symmetry
Odd functions –
have the origin as the point of symmetry.
*A graph has point symmetry if there is a ½ turn (rotation of 180 deg) that maps the graph onto itself.
Review point of symmetry
Axis of symmetry

12. Try these
Graph each function. Describe its symmetry. Tell whether it is even, odd, or neither.
Y = x5
Y = IxI
Y = 2x + 3
Y = 2x4
Even functions – have an even exponent, so they have the y-axis as the axis of symmetry.
Odd functions – has the origin as the point of symmetry (odd exponent)

13. The Degree of a Polynomial
Monomial
Binomial
Trinomial
Polynomial

14. Describing Polynomials
Degree
Name using degree
# of terms
Name using # terms 6
constant 1
monomial
X + 3
Linear 2
Binomial
3x2
Quadratic
Monomial
X3-5x2-3 3
Cubic
trinomial

15. Classify each polynomial
Name & degree:
5x2 – 7x
-10x3
X2 + 3x - 2
binomial, 2
Monomial, 3
Trinomial, 2

16. Assignment Section 9.1 p.560 – 562 #1 – 5, 11 – 14,
#16 – 20 (calculator)
#22 – 24 (calculator)

17. Warm-up 1/30/08 *Use stat, edit, then cubic reg:

18. 9.2: Graphs of Polynomials
LEQ: How do you find important points on a polynomial function? Vocabulary…

19. Important Points: Maximum Value Minimum Value Extreme Values (extrema)
Relative Extremum (turning points)
Relative Maximum
Relative Minimum
Zeros (roots) – using “table”

20. End Behavior Positive Intervals Negative Intervals Increasing Interval
Slope between two points is increasing
Decreasing Interval
Slope between two points is decreasing

21. Assignment
Section 9.2 P
#1-15

22. Warm-up 1/31/08 Solve the following system: d = 43 c – d =12
2b + 3c – 4d = 201
3a + b = -40

23. §9.3: Polynomial Models
How do you find a polynomial model for a given set of data? Given the points: (1,1), (2,4), (3,10), (4,20) Find the degree of a function that would best model the function.

24. The Tool:
Polynomial Difference Theorem: The function y = f(x) is a polynomial function of degree n iff the nth difference of corresponding y-values are equal and non-zero.

25. Example
F(x) = 5x + 3 Pick x’s to plug in: x y’s st diff: This function is of degree 1.

26. Example:
X’s f(x)=x2+x st difference nd difference This is a 2nd degree model.

27. Ex.
Given the pairs, determine what the degree of the function is: X’s Y’s st diff: nd diff: rd diff: Thus, a cubic model is best for the data. Use stat, Calc, cubic reg. to determine the fn. The function should be: f(x)= 2x3 + x - 10

28. Determine if y is a polynomial function of x of degree less than 5
Determine if y is a polynomial function of x of degree less than 5. IF so, find an equation of least degree for y in terms of x.

29. Assignment
Section 9.3 P #1,2,5-7, 9a-c 11a-c 15,16,20,21

30. Warm-up 2/4/08 Let f(x) = 3x2 – 40x + 48
Which of the given polynomials is a factor of f(x)?
x – 2 d) x - 6
x – 3 e) x – 12
x – 4 f) x – 24
Which of the given values equals 0?
f(2) c) f(4) e) f(12)
f(3) d) f(6) f) f(24)

31. Synthetic Division (dividing polynomials)
When you are dividing by a linear factor, you can use a simplified process (synthetic division).
Linear factor – graphed, it would be a line.
In synthetic division
-omit all variables & exponents
-reverse the sign of the divisor so you can add

32. Steps:
Reverse the sign of the divisor. Use 0 as a “place holder” for any missing term.
Bring down first coefficient.
Multiply the first coefficient by the divisor & write under next coefficient.
Add numbers, bring down.
Repeat steps 1-4 until done.

33. Examples (Synthetic Division)
Divide x2 + 3x – 12 by x – 2
x + 5 – 2/x – 2
2) Divide x2 + 5x + 6 by x + 2
x + 3
3) Divide 2x2 – 19x + 24 by x – 8.
2x - 3

34. 9.5: Polynomials & Linear Factors
Sometimes its more useful to work with polynomials in factored form:
Ex. X3 + 6x2 + 11x + 6 =
(x + 1)(x + 2)(x + 3)
How could you prove these are factors of the polynomial?

35. 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.
Ex. Graph y = x2 + 3x – 4
It crosses the x-axis at 1.
Thus, (x – 1) is a factor of the polynomial.
Also (x + 4) is a factor of the polynomial.

36. If a function is in factored form, you can use the zero product property to find the values that will make the function equal zero.
Ex. F(x) = (x – 1)(x + 2)(x – 4)
For what values of x will f(x) = 0?
1, -2, 4

37. What the answer tells you:
In the previous example, (x – 1) and (x + 4) were the factors of the polynomial.
Therefore:
1) -4 is a solution of x2 + 3x – 4
2) -4 is an x-intercept of y = x2 + 3x – 4
3) -4 is a zero of y = x2 + 3x – 4
4) x + 4 is a factor of y = x2 + 3x - 4

38. Factor-Solution-Intercept Equivalence Theorem
All the following are equivalent statements
(x – c) is a factor of f
f(c) = 0
C is an x-intercept of the graph y = f(x)
C is a zero of f
The remainder when f(x) is divided by
(x – c) is 0.

39. Ex) Determine the zeros of the function
y = (x – 2)(x + 1)(x + 3)
x = 2, -1, -3
Ex) Write four equivalent statements about one of the solutions of the equation
x2 – 4x + 3 = 0
Factors (x – 3)(x – 4)

40. Ex1 p.585 Factor g(x) = 6x3 – 25x2 – 31x + 30
Use a calculator/graph to see if any roots are obvious.
Verify the root (calculator or plug in)
Use the conjugate as a factor & complete long division
Continue factoring all terms of higher power than 1

41. Finding Polynomials with Known 0’s
Find an equation for a polynomial with zeros of -1, 4/5, and -8/3. Easy: (x + 1)(x – 4/5)(x + 8/3) = f(x) But, this is just one, to allow for other possibilities: If k is any non-zero term, this is more general: k(x + 1)(x – 4/5)(x + 8/3) = g(x)

42. Assignment
Section 9.5 P #2-13, 17-18,20

43. Warm-up 2/5/08 Find a value of k so that the graph of
P(x) = 2x4 – 5x2 + k
Intersects the x-axis in the indicated number of points. 1 4 3

44. Permanent Seats Want to move?
Preferences
Permanent Seats Want to move?

45. §9.6: Complex Numbers
Refresh Calculate (2 + 2i)4 Theorem If “a” and “b” are real numbers then a2 + b2 = (a + bi)(a – bi)

46. Factor into linear factors:
3p2 – 4
(√3p – 2 )(√3p + 2)
2) 4p2 – 9
(2p – 3)(2p + 3)
3) 4p2 + 9
(2p + 3i)(2p – 3i)

47. Word Problem
The volume of a container is modeled by the function V(x) = x3 – 3x2 – 4x.
Let x represent the width, x + 1 the length,
and x – 4 the height. If the container has a volume of 70 ft3, find its dimensions.
70 = x3 – 3x2 – 4x
5.78 x 6.78 x 1.78 ft

48. §9.7: The Fundamental Theorem of Algebra
LEQ: How is the Fundamental Theorem of Algebra used to factor polynomials? Introduction… In textbook Read p (through FToA)

49. Fundamental Theorem of Algebra
A polynomial function of degree “n” has exactly n zeros.
-Some of the zeros may be imaginary and some may be multiple zeros.
-You may have to use various methods to find all the answers:
Methods:
-graphing, factor theorem,
polynomial division, and quadratic formula

50. Zeros
IF “P” is a polynomial function of a degree n (greater than 1) with real coefficients, the graph can cross ANY horizontal line at most n times.
Ex) How many zeros does g(x) have?
g(x) = -3x5 – xi

51. Ex. What is the lowest possible degree of f(x)?

52. Ex. What is the lowest possible degree of f(x)?

53. Conjugate Zeros Theorem
Given a polynomial that has a complex root a + bi, it can be proved that a – bi is also a root of the polynomial.

54. Example 3 (p.599) Let p(x) = 2x3 – x2 + 18x – 9
Verify that 3i is a zero of p(x).
Verify by plugging into the function.
Find the other roots of the function.
-3i is a root (by conjugate zeros theorem)
Now find the last root by
-graphing, factoring, or division

55. Multiplicity
When a linear factor (zero point) is repeated, it is called a “multiple zero”.
Ex. Write a polynomial function with zeros at -2, 3, 3.
Factors : (x + 2)(x – 3)(x – 3)
Standard form: x3 – 4x2 – 3x + 18
Because (x – 3) is repeated 2x, the function has a multiplicity of 2 at 3.

56. What are the zeros and multiplicity?
Y = (x – 1)2(x -2)
multiplicity of 2 for 1
multiplicity of 1 for 2
Y = (x – 2)(x – 2)2
multiplicity of 1 for 1
multiplicity of 2 for 2

57. Write a polynomial function in standard form with the given zeros:
-1, 0, 4
factors: (x + 1)(x)(x – 4)
y = x3 – 3x2 – 4x
-2, -2, 5
factors: (x + 2)(x + 2)(x – 5)
y = x3 – x2 -16x - 20

58. Graph each of the following and note how the curve behaves at x = 2.
f(x) = (x – 2)(x + 1)(x + 3)
F(x) = (x – 2)2(x + 1)(x + 3)
F(x) = (x – 2)3(x + 1)(x + 3)
F(x) = (x – 2)4(x + 1)(x + 3)
What does the multiplicity of a real zero have to do with the way the graph behaves at the x-intercept?

59. Warm-up
For each function, determine the zeros and their multiplicity.
Y = (x – 2)2(x + 3)(x + 2)
Y = (3x – 2)4(2x + 2)
Write the following polynomial in standard form.
3) Y = (x + 3)2(x – 2)
After completing warm-up,
Go over homework assignment
Go over polynomials worksheet that was assigned last week!
Finding roots by using the graphing calculator.

60. Severe Weather Awareness Week
Reminders Expect a drill sometime this week Where will you go? What should you do? You should be COMPLETELY silent Drill is practice for a real event

61. Extra Credit Paper Famous Mathematician
Four pages
Double spaced
Size font
Must have 2+ reputable reference sources on a cited reference page (APA/MLA ok)
On a famous mathematician
Include background, what they’re famous for, & a typical problem they might work with explanations.
Must me an electronic copy
Ex) cphippen.pythagoras
Due Thursday, February 14th (half day)

62. Assignment
As a class p.600 – 602 #11, 16, 17, 18, 19, 20, 27 Homework p. 600 – 601 #1 – 2, 3 (in your own words), 4 – 14 (skip 11), 26

63. Quiz (Review)

64. Warm-up 2/7/08 Find all factors: x4 – 1 2x3 + 3x2 + x

65. §9.8: Factoring Sums and Differences
LEQ: How do you solve polynomial equations by factoring? Topics: Sums and Differences of Cubes Theorem Sums and Differences of Odd Powers Theorem (Even Powers – numerical analysis) EXAMPLES!!!

66. Assignment
Section 9.8 p #2,3,5-7,9-12,13-14a,20-23 Self Assessment p. 621 #1-16 (skip 14) *9, 11

67. Do what you need to do to excel
Homework
Write a note card
(theorems, formulas)
“Study Guide”
Do what you need to do to excel p
#2 -14, 27 – 30, 34, , ,

68. Warm-up 2/8/08
Factor x3 – 64y12 over the set of polynomials with rational coefficients.
2) Factor m6 + n6 over the set of polynomials with rational coefficients.
3) Factor x3 – 64 completely (over real and imaginary roots)
(x – 4y4)(x2 + 4xy4 + 16y8)
(m2 + n2)(m4 – m2n2 + n4)
(x – 4)(x2 + 4x + 16); roots: 4, -2 ± 2√3i