Warm-up1/28/08

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)(x1) 2x(x-1) 2x(x1)
• S.A. 6x2 2
• Volume lwh
• V x(x-1)(x1)
• V x3 - x

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.
• Power function
• 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.

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

13
The Degree of a Polynomial

• Monomial
• Binomial
• Trinomial
• Polynomial

14
Describing Polynomials

Polynomial Degree Name using degree of terms Name using terms
6 0 constant 1 monomial
X 3 1 Linear 2 Binomial
3x2 2 Quadratic 1 Monomial
X3-5x2-3 3 Cubic 3 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 1/30/08Use 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.567-568
• 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 xs to plug in
• x -2 -1 0 1 2 3 4
• ys -7 -2 3 8 13 18 23
• 1st diff 5 5 5 5 5 5
• This function is of degree 1.

26
Example

• Xs -1 0 1 2 3 4 5
• f(x)x2x3 3 3 5 9 15 23
  33
• 1st difference 0 2 4 6 8
  10
• 2nd difference 2 2 2 2 2
• This is a 2nd degree model.

27
Ex.

• Given the pairs, determine what the degree of the
  function is
• Xs 0 1 2 3 4
  5 6
• Ys -10 -7 8 47 122 245
  428
• 1st diff 3 15 39 75 123
  183
• 2nd diff 12 24 36 48
  60
• 3rd diff 12 12 12 12
• 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.IF so, find an equation of
least degree for y in terms of x.
29
Assignment

• Section 9.3
• P.547-576
• 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

1. Reverse the sign of the divisor. Use 0 as a
   place holder for any missing term.
2. Bring down first coefficient.
3. Multiply the first coefficient by the divisor
   write under next coefficient.
4. Add numbers, bring down.
5. 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 0s

• 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.587-588
• 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
• 0

44
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
• (v3p 2 )(v3p 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

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.596-597 (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 6.4

• 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)

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 PaperFamous Mathematician

• Four pages
• Double spaced
• Size 12-14 font
• Must have 2 reputable reference sources on a
  cited reference page (APA/MLA ok)
• On a famous mathematician
• Include background, what theyre famous for, a
  typical problem they might work with
  explanations.
• Must e-mail 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
• 2x3 2x2 x2 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. 605-606
• 2,3,5-7,9-12,13-14a,20-23
• Self Assessment
• p. 621
• 1-16 (skip 14)
• 9, 11

67
Homework

• Write a note card
• (theorems, formulas)
• Study Guide
• Do what you need to do to excel
• p. 622-624
• 2 -14, 27 30, 34, 36 - 39, 42 - 44, 48 - 53

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 2v3i