1. Unit 3: Lesson 1: 3.1 Polynomial Basics (4-1)
Lesson 2: Remainder Theorem (4-3)
Lesson 3: RRT (4-4,4-5)
Lesson 4: Solving Rational Equations (4-6)
Lesson 5: Solving Radical Equations (4-7)
Lesson 6: Modeling Real World Data (4-8)

2. Unit 3: “What The Function?”
In this unit we will cover…
Standard 3.1: determine the number and type of roots of a polynomial function or to build a polynomial function from its roots (4-1)
Standard 3.2: use the remainder and factor theorems to find roots of polynomial functions (4-3)
Standard 3.3: use the rational root theorem and rule of signs to find roots of polynomial functions (4-4, 4-5)
Standard 3.4: solve rational equations and inequalities (4-6)
Standard 3.5: solve radical equations and inequalities (4-7)

3. In this section we will…
Standard 3.1: determine the number and type of roots of a polynomial function or to build a polynomial function from its roots (4-1)
In this section we will…
Learn to recognize polynomial equations.
Find the degree and leading coefficient of a polynomial.
Determine roots of polynomial equations.
Apply the Fundamental Theorem of Algebra.

4. Polynomial Functions in One Variable:
Degree:
Equal to the highest power of the variable present.

5. Recognizing a Polynomial Function: State yes or no; if yes state degree.
No Yes, degree 5
Yes, degree 4 No

6. Roots: A polynomial of degree n has exactly…
Complex roots are real and imaginary roots in the form a + bi.
Ex. 2, 3i, 4+3i, 4-3i
Real roots appear on the graph of a function as _________________. Imaginary do not.
Another name for roots is _________.
Can you ever have just one imaginary root?
These pairs are called “complex conjugates”.

7. Roots: Let’s look at some scenarios: 1st degree 2nd degree 3rd degree
How many roots are possible?
How many combinations of real and imaginary roots can you have?
1st degree
2nd degree
3rd degree
4th degree
5th degree

8. Finding Roots of Polynomial Functions
State the number of complex roots of the equation x3 + 2x2 – 8x = 0. Then find the roots and graph the related function.

9. Corollary to the Fundamental Theorem of Algebra
Every polynomial P(x) of degree n (n > 0) can be written as the product of a constant k (k ≠ 0) and n linear factors.
Thus, a polynomial equation of degree n has exactly n complex roots.

10. Writing Polynomial Functions
Write the polynomial equation of least degree with roots -3, 2i, and -2i.

11. In this section we will…
Standard 3.1: determine the number and type of roots of a polynomial function or to build a polynomial function from its roots (4-2)
In this section we will…
Solve quadratic equations.
Use the discriminant to describe the roots of a quadratic equation.

12. Quadratic Equations:
A quadratic equation is simply a polynomial of __________ degree.
Their parent graph is a ________________.
To find its roots you can _______________, or you can __________________________,
or use the __________________________.

13. The Quadratic Formula:

14. The Discriminant: Discriminant: If the discriminant equals zero:
If the discriminant is positive:
If the discriminant is negative:

15. Completing the square Move number away from rest of quad.
Take ½ the middle coeffcient.
Square your answer.
Add it to quad and to the number.
Factor the trinomial (Use answer from step 2)

16. Completing the square…again
Move number to other side of equal sign.
Divide out the coeff. of g2.
Take ½ the middle coeffcient.
Square your answer.
Add it to quad and to the number.
Factor the trinomial. (Use answer from step 2)

17. Using the Quadratic Formula:
Find all roots of f(x) = 2w2 + 3w + 3 = 0. Then graph the function.

18. Using the Discriminant:
Determine the nature of the roots of
x2 – 3x – 7 = 0. Find the roots then sketch the graph based on them.

21. Homework: HW1 3.1: P 210 #15 – 49 odd, 55 HW2 3.1: P 219 #13 – 37 odd
Reassessments must be complete by Monday 3/14

22. Warm-Up: Draw the graph of a 4th degree polynomial that has:
No real roots
One real root
Two real roots
Three real roots
Four real roots
Five real roots

23. Homework:

24. In this section we will…
Standard 3.2: use the remainder and factor theorems to find roots of polynomial functions (4-3)
In this section we will…
Find the factors of polynomials using the Remainder and Factor Theorems.
Review synthetic division.

25. P(x) = (x – r)•Q(x)+P(r)
The Remainder Theorem
If a polynomial P(x) is divided by x – r, the remainder is a constant, P(r), and
P(x) = (x – r)•Q(x)+P(r)
where Q(x) is a polynomial with degree one less than the degree of P(x).

26. Huh?
Let’s check it out with numbers:

27. Okay…kinda Now with polynomials:
Let’s review the steps for synthetic division.

28. Synthetic Division:
Step 1: Arrange the terms of the polynomial in descending order. Hold the places of any missing powers of x with a zero. Write down the coefficients of each term (including zero’s).

29. Synthetic Division:
Step 2: Write the r (opposite sign because it is x – r) as the divisor in the box. Set up a line leaving space for numbers and another box to hold your remainder under the last coefficient.

30. Synthetic Division: Step 3: Bring down 1st coefficient.
Step 4: Multiply the 1st coefficient by r. Write the product under the 2nd coefficient. Add.
Step 5: Multiply the sum by r. Write the product under the next coefficient. Repeat for all coeff.’s
Step 6: The final sum in the box is the remainder. The other numbers represent Q(x), the quotient polynomial.

31. Hooray! We now know that…
But…The Remainder Theorem says: P(x) = (x – r)•Q(x)+P(r)
What gives?

32. Find P(r): In our case P(3)
Who was P(x)?
If this really works we will have 2 methods to find the remainder!

33. Let’s check it again! Let P(x)=2x3 + x2 – 4x + 3.
Check to see if the remainder found when P(x) is divided by (x+1) is the same as the value of P(-1).
It works! We can find the remainder with either method!

34. More Remainder Stuff
Find the remainder when P(x)=x3 + 3x2 – 2x – is divided by (x + 2) using both methods.

35. The Factor Theorem
The binomial x – r is a factor of the polynomial P(x) iff P(r) = 0.

36. 4-3 The Factor and Remainder Theorems
Determine whether x – 5 is a factor of P(x) if P(x) = x3 – 4x2 – 7x + 10.

37. Using the Factor Theorem:
Find all factors of f(x) = x3 + x2 – 17x + 15.
Answer: (x – 1)(x + 5)(x – 3)

38. Setting up a polynomial so that you get a certain root to be a factor:
Find a value for k so that (x+2) is a factor of g(x) = x3 + 8x2 + kx + 4.

39. Do another:
Find the value of j so that the remainder of (x3 + 6x2 – jx – 8) ÷ (x – 2) is 0.

40. Try this…
Determine how many times 1 is a root of x3 – 3x + 2 = 0.

42. Homework:
p 226 #15 – 39 odd and 43

43. Warm-Up:
Quiz!!!!

44. Homework:

45. In this section we will…
Standard 3.3: use the rational root theorem and rule of signs to find roots of polynomial functions (4-4)
In this section we will…
Use Descartes’ Rule of Signs to determine how many positive and negative roots a polynomial function will have.
Learn to identify all the possible rational roots of a polynomial function.
Test the possible roots in order to find a real root of the function.

46. Descartes’ Rule of Signs
Helps us find the number of possible positive real roots and the number of possible negative real roots.

47. Descartes’ Rule of Signs
If P(x) is a polynomial whose terms are arranged in descending powers of the variable, then..
The number of possible positive real roots =
The number of possible negative real roots =
The number of sign changes in the P(x)
The number of sign changes in the P(-x)

48. Descartes’ Rule of Signs
State the number of possible complex zeros, the number of possible positive real zeros, and the number of possible negative real zeros for:
h(x) = x4 – 2x3 + 7x2 + 4x – 15

49. Descartes’ Rule of Signs
Which would you look for first?
1 positive, 3 or 1 negative
2 or 0 positive, 1 negative
4 or 2 or 0 positive, 2 or 0 negative

50. The Rational Root Theorem
Consider a polynomial of degree n with integral coefficients:
If this polynomial has any rational roots, they will be in the form , where p is a factor of a0 and q is a factor of an.

51. Consider the function How many total complex roots does it have?
How many possible positives?
Possible negatives?
Determine the possible rational zeros.
Which of these possible rational zeros are actual zeros of the function?

52. The RRT and Rule of Signs x4 – 5x3 + 9x2 – 7x + 2 = 0
How many total complex roots does it have?
How many possible positives?
Possible negatives?
Determine the possible rational zeros.
Which of these possible rational zeros are actual zeros of the function?

53. Try another… How many total complex roots does it have?
How many possible positives?
Possible negatives?
Determine the possible rational zeros.
Which of these possible rational zeros are actual zeros of the function?

54. In this section we will…
Standard 3.3: use the rational root theorem and rule of signs to find roots of polynomial functions (4-5)
In this section we will…
Approximate the real zero’s of a polynomial function.

55. See the graph? See the approximate location of the zeros?
Look at the table. What happens near our approximate zeros?
That’s the Location Principle!

56. The Location Principle

57. Do we use this much?
Nope!
How do we approximate zeros? Look at the graph again.
Choose 2nd CALC.
Pick 2: ZERO
That’s how we do it.

58. How can it help?
Checking the calculator may give you a integral root (an integer).
Knowing one or more roots will help you to get the depressed polynomial and final factoring faster.

59. Homework: P 234 #11 –27 odd DO THEM LIKE THIS!
State the number of roots for the function (=degree).
Use Descartes’ Rule of Signs to state the number of possible positive and possible negative real roots.
Use RRT to find all the possible rational roots using p/q.
Use synthetic division to determine all the actual rational roots of the function.

60. Warm-Up:
P A32
Lesson 4-1: #1, 5, 9
Lesson 4-2: #3, 7

61. Homework:

62. Standard 3.4: solve rational equations and inequalities (4-6)
In this section we will…
Solve rational equations.
Solve rational inequalities.
We will NOT be finding partial fractions.

63. Rational Equations: First, find the domain of each part:
Solve the proportions by cross-multiplying.
Check answers against the known domain.

64. Find Domain
Determine LCD.
Build up each fraction so that they all have LCD.
Multiply by and remove LCD.
Solve.

65. Now You Try… Find Domain Determine LCD.
Build up each fraction so that they all have LCD.
Multiply by and remove LCD.
Solve.

66. Rational Inequalities:
Find domain.
Solve.
Set up solution(s) and points of discontinuity on number line.
Test a value in each region.
Final solution set.

67. Find domain.
Solve.
Set up solution(s) and points of discontinuity on number line.
Test a value in each region.
Final solution set.

68. Warm-up:

69. Homework:

70. Standard 3.5: solve radical equations and inequalities (4-7)
In this section we will…
Solve radical equations.
Solve radical inequalities.

71. Radical Equations: Radical means ANY root – not just square roots.
The big thing here is to CHECK YOUR ANSWERS in the original equation!
Radical equations like to throw false answers, so be careful.

72. Domain.
Isolate radical.
Raise both sides to power to remove radical.
Solve.
CHECK YOUR ANSWERS!
Could we see it coming?

73. Domain.
Isolate radical.
Raise both sides to power to remove radical.
Solve.
CHECK YOUR ANSWERS!

74. Domain.
Isolate radical
Uh Oh! Can’t isolate radical! So get one on each side.
Raise both sides to power to remove radical (Don’t forget to FOIL).
Isolate radical.
Raise both sides to power to remove radical
Solve.
CHECK YOUR ANSWERS!

75. Radical Inequalities:
Domain.
Isolate radical.
Raise each side to power to remove radical.
Solve.
Test solution(s) and domain restriction intervals.
Final solution set (includes any domain restrictions necessary).

76. Domain.
Isolate radical.
Raise each side to power to remove radical.
Solve.
Test solution(s) and domain restriction intervals.
Final solution set (includes any domain restrictions necessary).

77. Homework: P 247 #13-21 odd, 29-37 odd, 41 p 255 #13 – 39 odd
UNIT 3 TEST FRIDAY!!!!

78. Warm-up:
Pick up a couple pieces of graph paper for the lesson then do…
p 249 #40, 49, 51, 55 and 57

79. Homework:

80. Modeling Real-World Data with Polynomial Functions (4-8)
In this section we will…
Write polynomial functions to model real-world data
Use polynomial functions to interpret real-world data.

81. Getting ready… Go to STAT PLOT ( press 2nd then Y = ) Choose 1:Plot 1
Turn to ON, make sure the TYPE shows scatter plot. Xlist should show L1and Ylist should show L2. Mark should be on 1st one the boxed point.

82. Choosing a Mathematical Model
First graph your data and sketch a best fit function.
Decide which of our function parent graphs best fits the data.
Use the calculator to develop the regression equation.

83. Let’s Git’er Done.
You need to set the window to fit your data. Go to WINDOW change settings: Xmin=0, Xmax=20, Xscl=2; Ymin= -5, Ymax=10, Yscl=2.
Press STAT, choose 1:Edit…
In the L1 column enter the x-values; in the L2 column enter the f(x)-values.
Press GRAPH.
Press STAT, move to the CALC column, choose 6:CubicReg. Press ENTER twice.

84. The Project:

85. Things to Remember…
You may need to adjust your window.

86. Homework:
P 262 #9 – 21 odd
Unit 3 Test Tuesday!!!