1. Chapter 2 – Polynomial, Power, and Rational Functions
HW: Pg. 175 #7-16

2. 2.1- Linear and Quadratic Functions and Modeling
Polynomial Functions-
Let n be a nonnegative integer and let a0, a1, a2,…, an-1, an be real numbers with an≠0. The functions given by
f(x)=anxn + an-1xn-1+…+a2x2+a1x+a0
Is a polynomial function of degree n. The leading coefficient is an.
f(x)=0 is a polynomial function.
*it has no degree or leading coefficient.

3. Identify degree and leading coefficient for functions:
F(x) = 5x3-2x-3/4
G(x) = √(25x4+4x2)
H(x) = 4x-5+6x
K(x)=4x3+7x7

4. Polynomial Functions of No and Low Degree
Name
Form
Degree
Zero Function
F(x) = 0
Undefined
Constant Function
F(x)=a (a≠0)
Linear Function
F(x)=ax+b (a≠0) 1
Quadratic Function
F(x)=ax2+bx+c (a≠0) 2

5. Linear Functions F(x) = ax+b Slope-Intercept form of a line:
Find an equation for the linear function f such that f(-2) = 5 and f(3) = 6

6. Average Rate of Change
The average rate of change of a function y=f(x) between x=a and x=b, a≠b, is
[F(b)-F(a)]/[b-a]

7. Modeling Depreciation with a Linear Function
Weehawken High School bought a $50,000 building and for tax purposes are depreciating it $2000 per year over a 25-yr period using straight-line depreciation.
What is the rate of change of the value of the building?
Write an equation for the value v(t) of the building as a linear function of the time t since the building was placed in services.
Evaluate v(0) and v(16)
Solve v(t)=39,000

8. Characteristics of Linear Functions y=mx+b
Point of View
Characterization
Verbal
Polynomial of degree 1
Algebraic
F(x)=mx+b (m≠0)
Graphical
Slant line with slope m and y-intercept b
Analytical
Function with constant nonzero rate of change m, f is increasing if m>0, decreasing if m<0

9. Quadratic Functions and their graphs:
Sketch how to transform f(x)=x2 into:
G(x)=-(1/2)x2+3
H(x)=3(x+2) 2-1
If g(x) and h(x) and in the form f(x)=ax2+bx+c, what do you notice about g(x) and h(x) when a is a certain value (negative or positive)?

10. Finding the Vertex of a Quadratic Function:
f(x)=ax2+bx+c
We want to find the axis of symmetry, which is x=-b/(2a).
Then:
The graph of f is a parabola with vertex (x,y), where x=-b/(2a). If a>0, the parabola opens upward, and if a<0, it opens downward.

11. Use the vertex form of a quadratic function to find the vertex and axis of the graph of f(x)=8x+4x2+1:
x=-b/(2a)

12. Find the vertex of the following functions:
F(x)=3x2+5x-4
G(x)=4x2+12x+4
H(x)=6x2+9x+3
f(x)=5x2+10x+5

13. Vertex Form of a Quadratic Function:
Any Quadratic Function f(x)=ax2+bx+c, can be written in the vertex form:
F(x)=a(x-h)2+k
Where (h,k) is your vertex
h=-b/(2a) and k=is the y

14. Using Algebra to describe the graph of quadratic functions:
F(x)=3x2+12x f(x)=a(x-h)2+k
=3(x2+4x) Factor 3 from the x term
=3(x2+4x+() - () ) Prepare to complete the square.
=3(x2+4x+(2)2-(2)2) Complete the square.
=3(x2+4x+4)-3(4) Distribute the 3.
=3(x+2)2-1

15. Find vertex and axis, then rewrite functions in vertex form: f(x)=a(x-h)2+k
F(x)=3x2+5x-4
F(x)=8x-x2+3
G(x)=5x2+4-6x

16. Characteristics of Quadratic Functions: y=ax2+bx+c
Point of View
Characterization
Verbal
Polynomial of degree ___
Algebraic
F(x)=______________ (a≠0)
Graphical
a>0
a<0

17. 2.2 Power Functions With Modeling
HW: Pg.189 #1-10

18. Power function F(x)=k*xa
a is the power, k is the constant of variation
EXAMPLES:
Formulas
Power
Constant of Variation
C=2∏r 1 2∏
A=∏r2 2 ∏
F(x)=4x3
G(x)=1/2x6
H(x)=6x-2

19. What is the power and constant of variation for the following functions:
F(x) = ∛x
1/(x2)
What type of Polynomials are these functions? (HINT: count the terms)

20. Determine if the following functions are a power function Given that a,h,and c represent constants,, and for those that are, state the power and constant of variation:
6cx-5
h/x4
4∏r2
3*2x ax
7x8/9

21. 2.3 Polynomial Functions of Higher Degree
HW: Pg 203 #33-42e

22. Graph combinations of monomials:
F(x)=x3+x
G(x)=x3-x
H(x)=x4-x2
Find local extrema and zeros for each polynomial

23. Graph:
F(x)=2x3 F(x)=-x3 F(x)=-2x4 F(x)=4x4 What do you notice about the limits of each function?

24. Finding the zeros of a polynomial function:
F(x)=x3—2x2-15x
What do these zeros tell us about our graph?

25. SKETCH GRAPHS: F(x)=3x3 + 12x2 – 15x H(x)=x2 + 3x2 – 16
G(x)=9x3 - 3x2 – 2x
K(x)=2x3 - 8x2 + 8x
F(x)=6x2 + 18x – 24

26. 2.4 Real Zeros of Polynomial Functions
HW: Pg. 216 #1-6

27. Long Division
3587/ (3x3+5x2+8x+7)/(3x+2)

28. Division Algorithm for Polynomials
F(x) = d(x)*q(x)+r(x)
F(x) and d(x) are polynomials where q(x) is the quotient and r(x) is the remainder

29. Fraction Form: F(x)/d(x)=q(x)+r(x)/d(x)
(3x3+5x2+8x+7)/(3x+2)
Write (2x4+3x3-2)/(2x2+x+1) in fraction form

30. Special Case: d(x)=x-k
D(x)=x-k, degree is 1, so the remainder is a real number
Divide f(x)=3x2+7x-20 by:
(a) x (b) x (c) x+5

31. We can find the remainder without doing long division!
Remainder Theorem: If a polynomial f(x) is divided by x-k, then the remainder is r=f(k) Ex: (x2+3x+5)/(x-2) k=2 So, f(k)=f(2)=(2)2+3(2)+5=15=remainder

32. Lets test the Remainder Theorem with our previous example:
Divide f(x)=3x2+7x-20 by:
(a) x (b) x (c) x+5

33. PROVE: If d(x)=x-k, where f(x)=(x-k)q(x) + r
Then we can evaluate the polynomial f(x) at x=k:

34. Use the Remainder Theorem to find the remainder when f(x) is divided by x-k
F(x)=2x2-3x+1; k=2
F(x)=2x3+3x2+4x-7; k=2
F(x)=x3-x2+2x-1; k=-3

35. Synthetic Division
Now we can use this method to find both remainders and quotients for division by x-k, called synthetic division.
(2x3-3x2-5x-12)/(x-3)
K becomes zero of divisor

36. STEPS: 3 | 2 -3 -5 -12 _____________
* Since the leading coefficient of the dividend must be the leading coefficient , copy the first “2” into the first quotient position.
* Multiply the zero of the divisor (3) by the most recent coefficient of the quotient (2). Write the product above the line under the next term (-3).
* Add the next coefficient of the dividend to the product just found and record sum below the line in the same column.
* Repeat the “multiply” and “add” steps until the last row is completed.
3 |
_____________

37. Use synthetic division to solve:
(x3-5x2+3x-2)/(x+1)
(9x3+7x2-3x)/(x-10)
(5x4-3x+1)/(4-x)

38. Rational Zero Theorem
Suppose f is a polynomial function of degree n1 of the form f(x)=anxn+…+a0
with every coefficient an integer. If x=p/q is a rational zero of f, where p and q have no common integer factors other than 1, then
P is an integer factor of the constant coefficient a0, and
Q is an integer factor of the leading coefficient an.
Example: Find rational zeros of f(x)=x3-3x2+1

39. Finding the rational zeros:
F(x)=3x3+4x2-5x-2
Potential Rational Zeros:

40. Find rational zeros:
F(x)=6x3-5x-1
F(x)=2x3-x2-9x+9

41. Upper and Lower Bound Tests for Real Zeros
Let f be a polynomial function of degree n≥1 with a positive leading coefficient. Suppose f(x) is divided by x-k using synthetic division.
If k≥0 and every number in the last line is nonnegative (positive or zero), then k is an upper bound for the real zeros of f.
If k≤0 and the numbers in the last line are alternately nonnegative and nonpositive, then k is a lower bound for the real zeros of f.

42. Example:
Lets establish that all the real zeros of f(x)=2x4-7x3-8x2+14x+8 must lie in the interval [-2,5]

43. Now we want to find the real zeros of the polynomial function f(x)=2x4-7x3-8x2+14x+8

44. Steps to finding the real zeros of a polynomial function:
Establish bounds for real zeros
Find the real zeros of a polynomial functions by using the rational zeros theorem to find potential rational zeros
Use synthetic division to see which potential rational zeros are a real zero
Complete the factoring of f(x) by using synthetic division again or factor.

45. Find the real zeros of a polynomial function:
F(x)=10x5-3x2+x-6

46. Find the real zeros of a polynomial function:
F(x)=2x3-3x2-4x+6
F(x)=x3+x2-8x-6
F(x)=x4-3x3-6x2+6x+8
F(x)=2x4-7x3-2x2-7x-4

47. 2.5 Complex Numbers

48. F(x)=x2+1 has no real zeros
In the 17th century, mathematicians extended the definition of √(a) to include negative real numbers a.
i =√(-1) is defined as a solution of (i )2 +1=0
For any negative real number √(a) = √|a|*i

49. Complex Number- is any number written in the form:
a +bi , where a, b are real numbers
a+bi is in standard form

50. Sum and Difference Sum: (a+bi) +(c+di) = (a+c) + (b+d)i
Difference: (a+bi) – (c+di) = (a-c) + (b-d)I
EX: (a) (8 - 2i) + (5 + 4i)
(b) (4 – i) – (5 + 2i)

51. Multiply:
(2+4i)(5-i)
Z=(1/2)+(√3/2)i, find Z2

52. Def: Complex Conjugates of the Complex Number z=a+bi is
Z = a+bi = a – bi
When do we need to use conjugates?

53. Write in Standard Form:
(2+3i)/(1-5i)

54. Complex Solutions of Quadratic Equations
ax2+bx+c=0

55. Solve: x2+x+1=0
Try: x2-5x+11=0

56. DO NOW:
Find all zeros:
f(x) = x4 + x3 + x2 + 3x - 6

57. 2.6 Complex Zeros and The Fundamental Theorem of Algebra
HW: Pg #2-10e, 28-34e

58. Two Major Theorems
Fundamental Theorem of Algebra – A polynomial function of degree n has n complex zeros (real and nonreal).
Linear Factorization Theorem – If f(x) is a polynomial function of degree n>0, then f(x) has n linear factors and
F(x) = a(x-z1)(x-z2)…(x-zn)
Where a is the leading coefficient of f(x) and z1, z2, …, zn are the complex zeros of the function.

59. Fundamental Polynomial Connections in the Complex Case
X=k is a…
K is a
Factor of f(x):

60. Exploring Fundamental Polynomial Connections Write the polynomial function in standard form and identify the zeros :
F(x)=(x-2i)(x+2i)
F(x)=(x-3)(x-3)(x-i)(x+i)

61. Complex Conjugate Zeros
Suppose that f(x) is a polynomial function with real coefficients. If a+bi is a zero of f(x), then the complex conjugate a-bi is also a zero of f(x)

62. EXPLORATION (with your partner):
What can happen if the coefficients are not real?
Use substitution to verify that x=2i and x=-i are zeros of f(x)=x2-ix+2. Are the conjugates of 2i and –i also zeros of f(x)?
Use substitution to verify that x=i and x=1-i are zeros of g(x)=x2-x+(1+i). Are the conjugates of i and 1-i also zeros of g(x)?
What conclusions can you draw from parts 1 and 2? Do your results contradict the theorem about complex conjugates?

63. Find a Polynomial from Given Zeros
Given that -3, 4, and 2-i are zeros, find the polynomial:
Given 1, 1+2i, 1-i, find the polynomial:

64. Find Complex Zeros of f(x)=x5-3x4-5x3+5x2-6x+8

65. Find Complex Zeros
The complex number z=1-2i is a zero of f(x)=4x4+17x2+14x+65, find the remaining zeros, and write it in its linear factorization.

66. Find zeros:
3x5-2x4+6x3-4x2-24x+16

67. 2.7 Graphs of Rational Functions
HW: Pg. 246 #19-30

68. Rational Functions
F and g are polynomial functions with g(x)≠0. the functions:
R(x)=f(x)/g(x) is a rational function
Find the domain of : f(x)=1/(x+2)

69. Transformations: Describe how the graph of the given function can be obtained by transforming the graph f(x)=1/x
g(x)=2/(x+3)
H(x)=(3x-7)/(x-2)

70. Finding Asymptotes
Find horizontal and vertical asymptotes of f(x)=(x2+2)/(x2+1)
Find asymptotes and intercepts of the function f(x)=x3/(x2-9)

71. Analyzing: f(x)=(2x2-2)/(x2-4)

72. Lets look at F(x)=(x3-3x2+3x+1)/(x-1)

73. 2.8 Solving Equations in One Variable
HW: Pg. 254 #7-17

74. Solve:
X + 3/x = 4
2/(x-1) + x = 5

75. Eliminating Extraneous Solutions:
(2x)/(x-1) + 1/(x-3) = 2/(x2-4x+3)
3x/(x+2) + 2/(x-1) = 5/(x2+x-2)

76. Try:
(x-3)/x + 3/(x+2) + 6/(x2 +2x) = 0

77. Finding a Minimum Perimeter:
Find the dimensions of the rectangle with minimum perimeter if its area is 200 square meters. Find the least perimeter:
A = 200

78. 2.9 Solving Inequalities in One Variable
HW: Finish 2.9 WKSH

79. Finding where a polynomial is zero, positive, or negative
F(x)=(x+3)(x2+1)(x-4)2
Determine the real number values of x that cause the function to be zero, positive, or negative:

80. Find solutions to: (x+3)(x2+1)(x-4)2 > 0 (x+3)(x2+1)(x-4)2 ≥ 0

81. Solving a Polynomial Inequality Analytically:
2x3-7x2-10x+24>0
Solve Graphically: x3-6x2≤2-8x
*Plug function into your calculator*

82. Practice:
Section 2.9 #1-12 odd
Check yourself