1. SPONGE
(3x + 2)4
(2x – 5y)3

2. Dividing Polynomials

3. Activating…. Take notes.
cedAlgebra/OperationsWithPolynomials/GSEAdvancedAl gebra_OperationswithPolynomials_Shared2.html#

4. Let’s try!

5. Simple Division -
dividing a polynomial by a monomial

6. Simplify

7. Simplify

8. COPY! How do you divide Polynomials using Long Division?
Divide the leading coefficients.
Multiply by the distributive property & align your terms.
Subtract the polynomials.
Bring down the next term.
Repeat the process for all terms.

9. Dividend
The answer = “Quotient “
Divisor

10. x
- 8
x2/x = x
-8x/x = -8
-( )
+ 3x x2
- 8x
- 24
-( )
- 8x
- 24

12. Your turn.

13. Your turn again.

14. Your turn again!

15. After lunch ….in case you forgot!
cedAlgebra/OperationsWithPolynomials/GSEAdvancedAl gebra_OperationswithPolynomials_Shared2.html#

16. Think Pair and Share… Your turn!

17. 2. Prepare for your quiz (20min only)
Sponge 1.
2. Prepare for your quiz (20min only)

18. When you are done with your quiz…..
Turn them in at my desk at get a laptop to begin on your Khan Academy..
Assignments : Polynomials Intro
2. Add polynomials Intro
3. Expand Binomials

19.
Synthetic
Division

20. Sponge Find the volume.

21. Check !

22. What is Synthetic Division?
cedAlgebra/OperationsWithPolynomials/GSEAdvancedAl gebra_OperationswithPolynomials_Shared2.html#

23. Copy: In order to use Synthetic Division….
There must be a coefficient for every possible power of the variable
The divisor must have a leading coefficient of 1.

24. Let’s look at how to do this using the example: YOU TRY!
Recall: In order to use synthetic division these two things must happen: #1
There must be a coefficient for every possible power of the variable. #2
The divisor must have a leading coefficient of 1.

25. Step #1: Write the terms of the polynomial so
Step #1: Write the terms of the polynomial so the degrees are in descending order.

26. Step #2: Write the constant a of the divisor
Step #2: Write the constant a of the divisor x- a to the left and write down the coefficients.
a= what goes here!!
From the divisor (opposite)
Coefficients

27. Step #3: Bring down the first coefficient, 5.
Step #4: Multiply the first coefficient by r (3*5).

28. Step #5: After multiplying in the diagonals, add the column.

29. Add
Multiply the diagonals, add the columns.

30. Add Columns Add Columns Add Columns Add Columns
Step #7: Repeat the same procedure as step #6.
Add Columns
Add Columns
Add Columns
Add Columns

31. Step #8: Write the quotient (answer).
The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

32. The quotient is:
Remember to place the remainder over the divisor.

33. Example 2

34. Check

35. Example 3.

36. Ex 3:
Step#1: Powers are all accounted for and in descending order.
Step#2: Identify r in the divisor.
Since the divisor is x+4, r=-4 .

37. Test review

38. 4 -4 20 8 -5 -1 1 -2 10 Step#3: Bring down the 1st coefficient.
Step#4: Multiply and add.
Step#5: Repeat. 4 -4 20 8 -5 -1 1 -2 10

39. Synthetic Division
Can be used when dividing by x – r term, where r is a number.
(4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0
2 |
60 is the reminder; 26 is the constant, 13 the 1st degree term, 4 the 2nd degree term

40. Synthetic Division
Can be used when dividing by x – r term, where r is a number.
(4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0
2 |
4x2 + 13x

41. The Factor Theorem
We use the factor theorem to help us find all “factors” of a polynomial.
If the remainder equals zero, the divisor is one of the factors.
To find the others you

42. 2 7 -4 -27 -18 +2 4 22 18 36 9 2 11 18 The Factor Theorem
A polynomial f(x) has a factor (x – k) if and only if f(k) = 0.
Show that (x – 2) and (x + 3) are factors of
f(x) = 2x4 + 7x3 – 4x2 – 27x – 18 2 7 -4
-27
-18 +2 4 22 18 36 9 2 11 18

43. Synthetic Division - To use synthetic division:
divide a polynomial by a polynomial
To use synthetic division:
There must be a coefficient for every possible power of the variable.
The divisor must have a leading coefficient of 1.

44. Step #1: Write the terms of the. polynomial so the degrees are in
Step #1: Write the terms of the polynomial so the degrees are in descending order.
Since the numerator does not contain all the powers of x, you must include a 0 for the

45. 5 -4 1 6 Since the divisor is x-3, r=3
Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients. 5 -4 1 6
Since the divisor is x-3, r=3

46. Step #3: Bring down the first coefficient, 5.

47. 15 15 5 Step #4: Multiply the first coefficient by r, so
and place under the second coefficient then add. 5 15 15

48. Step #5: Repeat process multiplying. the sum, 15, by r;
Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add. 5 15 45 41

49. Step #5 cont.: Repeat the same procedure.
Where did 123 and 372 come from? 5 15 45 41
123
372
124
378

50. Step #6: Write the quotient.
The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend. 5 15 45 41
123
124
372
378

51. The quotient is:
Remember to place the remainder over the divisor.

52. Ex 8:
Notice the leading coefficient of the divisor is 2 not 1.
We must divide everything by 2 to change the coefficient to a 1.

53. 3

54. *Remember we cannot have complex fractions - we must simplify.

55. Ex 9: 1
Coefficients

57. Practice

58. SPONGE

59. Sponge Do not turn in HW at this time.

60. HW CHECK! Volunteers!

61. Review and take notes.
cedAlgebra/OperationsWithPolynomials/GSEAdvancedAl gebra_OperationswithPolynomials_Shared2.html#

62. Class work practice! Handout 1

63. SPONGE- simplify

64. Function Composition
UNIT 2

65. Relation: Any set of ___________ that has an _____________.
Notes
Relation: Any set of ___________ that has an _____________.
Function: A _________________ such that every single ___________ has exactly ___________ output.

66. Function Notation: Function notation is a way to ________________.
It is pronounced ________________________.
f(x) is a fancy way of writing _____ in an _______________.
Example: y = 2x + 4 is the same as
___________________

67. Examples 1-3

68. Ex 2.

70. Given

71. Function Composition Given g(x) = 3x - 4x2 + 2, find g(5)
15 - 4(25) + 2 =
= -83
g(5) = -83

72. SPONGE 1. Given g(x) = 2x2 - 3x+5, find g(-3)

73. Function Composition
Function Composition is just fancy substitution, very similar to what we have been doing with finding the value of a function.
Key point! We will be plugging one function into the other function

74. The notation looks like g(f(x)) or f(g(x)).
Notation Used….
The notation looks like g(f(x)) or f(g(x)).
We read it ‘g of f of x’ or ‘f of g of x’

75. EX1. The COMPOSITION of the function f with g is
Plug the second function into the first

76. EXAMPLE 2
Given f(x) = 2x + 2 and g(x) = 2, find f(g(x)).

77. Example 3
Given g(x) = x - 5 and f(x) = x + 1, find f(g(x)).

78. Instructional Video - RECAP

79. Example 4 YOUR TURN Find g(f(x)) Given g(x) = x - 5 and f(x) = x + 1

80. Example 4b YOUR TURN now find…
Example 4b YOUR TURN now find…. Find f(g(x)) Given g(x) = x - 5 and f(x) = x + 1

81. 3RD BLOCK CLASSWORK 1, 3, 5, AND 7 “COMPOSITION”
You may collaborate with your peers! Must show work on a separate sheet of paper.
Copy problem and show work for all 4.

82. Function Inverses- AFTER LUNCH

83. COPY: Function Inverses
If given a relation set!
Simply switch the x and y values! That’s the inverse when given a relation!
If given an equation or function with f(x)
Simply switch the x and y in the equation and then solve for the new “Y”

84. Function Inverse {(2, 3), (5, 0), (-2, 4), (3, 3)} Inverse
Inverse = switch the x and y, (domain and range)
Inverse
= {(3, 2), (0, 5), (4, -2), (3, 3)}

85. {(4, 7), (1, 4), (9, 11), (-2, -1)} Inverse = ?
Ex2. Function Inverse
{(4, 7), (1, 4), (9, 11), (-2, -1)}
Inverse = ?
I = {(7, 4), (4, 1), (11, 9), (- 1, -2)}

86. Function Inverse for equations!
Switch the x and y in the equation and then solve for the new “Y”

87. Function Inverse Ex1 given f(x) = x + 5. 1st Substitute y for f(x).
y = x + 5.
Then switch x and y.
x = y + 5
Now solve for y.
x - 5 = y (this is your inverse equation)

88. Final answer!
So the inverse of f(x) = x + 5 is f-1(x) = x - 5 with correct notation!

89. Ex2. Given f(x) = 3x - 4, find its inverse (f-1(x)). y = 3x – 4
1st switch x and y. x = 3y - 4
2nd solve for y.

90. Your turn! Given h(x) = -3x + 9, find it’s inverse. y = -3x + 9
x = -3y + 9
x - 9 = -3y
(x - 9) / -3 = f-1(x)

91. Classwork “Composition” Handout #2, 3 ,4 and 6
You may collaborate with your peers! Must show work on a separate sheet of paper.
Copy problem and show work for all 4.

92. Function Inverse
Given
Find the inverse.

93. Function Inverse

94. Function Inverse
3x = 2y + 5
3x - 5 = 2y

95. Function Inverse
Given f(x) = x2 - 4
y = x2 - 4
x = y2 - 4
x + 4 = y2

96. Function Inverse
x + 4 = y2