1. Properties of Exponents
Product of Powers
Power of a Power
Power of a Product
Negative Exponents
Zero Exponent
Quotient of Powers
Power of a quotient

2. Examples

3. Polynomial Functions -exponents are whole numbers -coefficients are real numbers
-2 is the leading coefficient
4 is the degree ( the highest exponent)
-7 is the constant term

4. The degree of a polynomial function is the exponent of the leading term when it is in standard form
Degree type
0 Constant
1 Linear
2 Quadratic
3 Cubic
4 Quartic

5. You can evaluate polynomial functions using -direct substitution -synthetic substitution

6. EVALUATING A FUNCTION given a value for x
DIRECT SUBSTITUTION:
- replace each x with the given value
- evaluate expression, following PEMDAS
Example: f(x) = 2x⁴ - 8x² + 5x – 7, for x = 3
2(3)⁴ - 8(3)² + 5(3) - 7
2(81) – 8(9) + 5(3) - 7
162 – 98

7. The solution is the last number written.
SYNTHETIC SUBSTITUTION:
- write polynomial expression in standard form (include all degree terms)
- write only coefficients (including zeros)
-use the given value of x in the process below
Example: f(x) = 2x⁴ - 8x² + 5x – 7, for x = 3
2x⁴ + 0x³- 8x² + 5x – 7
X = 3 6 18 10 30 35
105 98 2
The solution is the last number written.

8. Find the recharge time after 100 flashes.
The time t ( in seconds)it takes a camera battery to recharge after flashing n times can be modeled by:
Find the recharge time after 100 flashes.
11.3 seconds

9. End Behavior of Polynomial Functions
is determined by the degree (n) and leading coefficient (a)
Use your graphing calculator to investigate the end behavior of several polynomial functions. Write a paragraph to explain how the leading coefficient and degree of the function affect the end behavior of these graphs
For a>0 and n even
For a>0 and n odd
For a<0 and n even
For a <0 and n odd

10. END BEHAVIORS WHAT THE GRAPH DOES AT THE ENDS?
If the degree is even the ends both go in the same direction
-If the leading coefficient is positive they both go up
-If the leading coefficient is negative they both go down
If the degree is odd the ends go in opposite directions
-if the leading coefficient is positive it’s climbing the stairs( going up from left to right)
-If the leading coefficient is negative it’s going down the stairs ( going down from left to right)

11. Leading Coefficient positive
Degreeeven
Leading Coefficient positive
End behavior of the function
Graph of the function
Example: f (x) = x2

12. Degree even
Negative
Leading
coefficient
Example: f (x) = –x2

13. Degree Odd
Positive
Leading
coefficient
Example: f (x) = x3

14. Degree Odd
Negative
Leading coefficient
Example: f (x) = –x3

15. POLYNOMIAL GRAPHS IT’S A MATTER OF DEGREES
MAX # OF ZEROS
MAX
TURNING POINTS
DEGREE/TYPE
EXAMPLE
END BEHAVIORS
0 /Constant y = Horizontal or infinity 0
1/linear y = -2x Alternate
2/quadratic y = x2 + 2x – Same
3/cubic y = x3 – 3x Alternate
4/quartic y = x4 – 4x3 – x2 + 12x – Same
n (odd) Alternate n n - 1
n (even) Same n n - 1

16. 6.3 OPERATIONS ON POLYNOMIALS
ADDITION:
Aka: combine like terms
EXAMPLE: (3 𝑥 𝑥 2 − 𝑥−7) + (𝑥 3 −10 𝑥 2 + 8)
Horizontally:
Vertically:
SUBTRACTION: add the opposite of the second polynomial
EXAMPLE: (3 𝑥 𝑥 2 − 𝑥−7) − (𝑥 3 −10 𝑥 2 + 8)
(3 𝑥 𝑥 2 − 𝑥−7) + (−𝑥 𝑥 2 − 8)

17. MULTIPLY
EXAMPLE: (𝑥 −3)(3 𝑥 2 −2𝑥 −4)
Horizontally:
Vertically:

18. APPLICATIONS OF POLYNOMIAL FUNCTIONS
From 1985 through 1995, the gross farm income G, and farm expenses, E (in billions of dollars), in the United States can be modeled by
G(t) = −.246 𝑡 𝑡 and E(t) = .174 𝑡 𝑡+131
Where t is the number of years since Write a model for the net farm income, N, for those years
N(t) = G(t) - E(t)
N(t) = (−.246 𝑡 𝑡+159) - (.174 𝑡 𝑡+131)
N(t) = −.42 𝑡 𝑡+28

19. APPLICATION: BOOK BUSINESS
From 1982 through 1995, the number of softbound books, N (in millions) sold in the United States, and the average price per book, P (in dollars) can be modeled by
𝑁(𝑡)=1.36 𝑡 𝑡 𝑎𝑛𝑑 𝑃(𝑡)= .314𝑡+3.42
Where t is the number of years since Write a model for the total revenue, R received from the sales of softbound books.
R(t) = P(t) x N(t)
𝑅(𝑡)= 𝑡 𝑡 𝑡
What was the total revenue from softbound books in 1990?
Method #1: evaluate R with t = 8
Method #2: graph R and determine R(8)
$7020 million ($7.02 billion)

20. Use synthetic substitution to evaluate for x=-2
After vacation warm up
Simplify
Write in standard form
Graph
Use synthetic substitution to evaluate
for x=-2

21. SPECIAL PRODUCT PATTERNS
SUM x DIFFERENCE: (a + b)(a – b) = a² - b²
Example: (x + 4)(x – 4) = x² - 16
SQUARE OF A BINOMIAL: (a + b)² = a² + 2ab + b²
Example: (x + 4)² = x² + 8x + 16
NOTE: The square of a binomial is always a trinomial.
CUBE OF A BINOMIAL: (a + b)³ = a³ + 3a²b + 3ab² + b³
Example: (x + 5)³ = a³ + 3a²b + 3ab² + b³
x³ + 3x²b(5) + 3x(25)² + 125
x³ + 15x² + 75x + 125

22. FACTORING REVIEW COMMON FACTOR: 6x² + 15x + 27 = 3( )
TRINOMIAL: 2x² -5x – 12 = ( )( )
PERFECT SQUARE TRINOMIAL: x² + 20x = ( )( )
DIFFERENCE OF TWO SQUARES: x² = ( )( )

23. MORE SPECIAL FACTORING PATTERNS
SUM OF 2 CUBES: a³ + b³ = (a + b)(a² - ab + b²)
Example: x³ = (x + 3)(x² - x(3) + 9
(x + 3)(x² - 3x + 9)
DIFFERENCE OF 2 CUBES: a³ - b³ = (a - b)(a² + ab + b²)
Try these:
x³ - 125
x³ + 64
27x³ - 8
343x³

24. Warm-up
Factor

25. ZERO PRODUCT RULE (STILL GOOD!)
Solving polynomial equations:
1. Transform equation to make one side zero
2. Factor other side completely
3. Determine values to make each factor zero
Example: 2x⁵ + 24x = 14x³
2x⁵ - 14x³ + 24x = 0
2x(x⁴ - 7x² + 12) = 0
2x(x² - 3)(x² - 4) = 0
2x(x² - 3)(x - 2)(x + 2) = 0
Set each factor to zero: 2x = 0 x² - 3 = x – 2 = 0 x + 2 = 0
x = x = ±√ x = x = -2

26. 𝑋 3 +27=0
(X + 3)(X² – 3X + 9) = 0
X + 3 = 0 OR
X² – 3X + 9 = 0

27. FACTOR BY GROUPING Use for polynomials with 4 terms
𝑟 3 −3 𝑟 2 +6𝑟−18
Separate into 2 binomials: 𝑟 3 −3 𝑟 𝑟−18
Factor out GCF of each:
𝑟 2 (𝑟−3)
+ 6(𝑟−3)
Factor out new GCF:
(𝑟−3)( 𝑟 2 +6)
TRY THESE:
𝑋 3 +6 𝑋 2 +7𝑋+42
𝑧 3 −2 𝑧 2 −16𝑧+32
25 𝑝 3 −25 𝑝 2 −𝑝+1
9 𝑚 𝑚 2 −4𝑚−8
CHECK:
(X² + 7)(X + 6)
(z² - 16)(z – 2)
(5p - 1)(5p + 1)(p – 1)
(3m - 2)(3m + 2)(m + 2)

28. Suppose you have 250 cubic inches of clay with which to make a rectangular prism for a sculpture. If you want the height and width each to be 5 inches less than the length, what should the dimensions of the prism be? Solve by factoring.

29. You are building a bin to hold cedar mulch for your garden
You are building a bin to hold cedar mulch for your garden. The bin will hold 162 cubic feet of mulch. The dimensions of the bin are x feet by 5x-9 feet by 5x-6 feet. How tall will the bin be?

30. In 1980 archeologists at the ruins of Caesara discovered a huge hydraulic concrete block with a volume of 330 cubic yards. The blocks dimensions are x yards high by (13x – 11) yards long by
(13x – 15) yards wide. What are the dimensions of the block?

31. You are building a bin to hold cedar mulch for your garden
You are building a bin to hold cedar mulch for your garden. The bin will hold 162 cubic feet of mulch. The dimensions of the bin are x ft. by (5x-6)ft. by (5x-9) ft. How tall will the bin be?

32. LONG DIVISION REVIEW
32040 /15 2 1 3 6
LONG DIVISION - Remember 4th grade? 15
Write dividend “inside the house”
3 0
Divide 1st digit(s) in dividend by the divisor; write answer in quotient 2
Multiply
1 5 5 4
Subtract
4 5
Bring down next digit 9
Repeat process as needed
9 0
Answer: 2136

33. POLYNOMIAL DIVISION (2x⁴ + 3x³ + 5x – 1) /(x² - 2x + 2) 2x² +7x + 10
LONG DIVISION
X² - 2x + 2
2x⁴ + 3x³ + 0x² + 5x – 1
Write dividend in standard form
(include all degrees)
2x⁴ - 4x³ + 4x²
Divide 1st term in dividend by
1st term in divisor
7x³ - 4x²
+5x
Multiply
7x³ - 14x² + 14x
Subtract
10x² -9x -1
10x² - 20x + 20
Bring down next term
11x - 21
Repeat process as needed
11x – 21
X² - 2x + 2
Answer: 2x² + 7x + 10

34. TRY THESE: Divide x² + 6x + 8 by x + 4 SOLUTIONS: X + 2
Verify: (x + 4)(x + 2)
Divide x² + 3x – 12 by x – 3
X + 6 R 6 or x 𝑥−3
Verify: (x – 3)(x + 6) + 6

35. Quotient: 1x2 + 3x + 2 SYNTHETIC DIVISION:
To divide polynomial f(x) by (x – k),
- write polynomial expression in standard form (include all degree terms)
- write only coefficients (including zeros)
- use the given value of k in the process below *
- The remainder is the last number written
- the other numbers in the answer are the coefficients/constant of the quotient
Example: Divide x3 - 3x2 - 16x – 12 , by ( x – 6)
x3 - 3x2 - 16x – 12
K = 6 6 3 18 2 12 1
Quotient: 1x2 + 3x + 2

36. TRY THESE: Divide x² + 6x + 8 by x + 4 SOLUTIONS: X + 2
Verify: (x + 4)(x + 2)
Divide x² + 3x – 12 by x – 3
X + 6 R 6 or x 𝑥−3
Verify: (x – 3)(x + 6) + 6

37. RELATED THEOREMS
REMAINDER THEOREM: If a polynomial f(x) is divided by x – k, then the remainder, r, equals f(k).
Remember synthetic substitution?
Example: (x3 + 2x2 – 6x – 9) ⁄ (x – 2)
K = 2 2 8 4 1 4 2 -5
f(2) = -5

38. Use the Remainder Theorem to evaluate P(-4) for P(x) = 2x4 + 6x3 – 5x2 - 60
WORKOUT -4
P(-4) = -12

39. SPECIAL CASE Use the Remainder Theorem to evaluate P(-3) for
P(x) = 2x3 + 11x2 + 18x + 9
- 3 2
Since P(-3) = 0:
1. 3 is a zero of P(x)
2. (x - ¯3) is a factor
(x + 3)
Quotient: 2x2 + 5x + 3
Note: the quotient is also factorable:
2x2 + 5x = (2x + 3) (x + 1)
Therefore, 2x3 + 11x2 + 18x = (x + 3) (2x + 3) (x + 1)
Try: if one zero is 2

40. OBSERVATION
When dividing f(x) by (x-k), if the remainder is 0, then (x – k) is __ ____________ of f(x).
Determine whether each divisor is a factor of each dividend:
a) (2x2 – 19x + 24) ÷(𝑥 −8) b) (x3 – 4x2 + 3x + 2) ÷ (x + 2)
yes no

41. A polynomial f(x) has a factor (x - k) if and only if f(k) = 0.
FACTOR THEOREM:
A polynomial f(x) has a factor (x - k) if and only if f(k) = 0.
Factor f(x) = 2x3 + 7x2 - 33x – 18 given that f(-6) = 0 -6
f(-6) = 0, so (x + 6) is a factor
Quotient: 2x2 – 5x -3
(which is the other factor, and can be factored into (2x + 1) (x – 3)
Therefore, 2x³ + 7x² - 33x – 18 = (x + 6)(2x + 1)(x – 3)

42. TRY THIS:
Given one zero of the polynomial function, find the other zeros.
F(x) = 15x3 – 119x2 – 10x + 16; 8
Since 8 is a zero, (x – 8) is a factor.
Since the quotient is 15x2 + x -2, it is also a factor.
Since 15x2 + x -2 can be factored into (5x + 2) (3x - 1).
The factors of 15x3 – 119x2 – 10x + 16 are
(x – 8) (5x + 2) (3x – 1)

43. Warm-up
Divide using long division.

44. The volume of a box is represented by the function
The volume of a box is represented by the function The box is (x-4) high and (2x+1) wide. Find the length.
V=lwh

45. WRITING A FUNCTION GIVEN THE ZEROS
Given: 2 and 4 are the zeros of the function f(x).
Write the function
f(x) = (x – 2) (x – 4)
f(x) = x2 – 6x + 8
Given: 3 and -4 and 1 are the zeros of the function f(x).
Write the function
f(x) = (x – 3) (x + 4) (x – 1)
f(x) = (x2 + x – 12) (x – 1)
f(x) = x3 – 13x + 12

46. Given the zeros of a function, write the function.
Try these:
Given the zeros of a function, write the function.
SOLUTIONS:
f (x) = x3 - 6x2 + 5x + 12
f (x) = x3 – 8x2 – 23x + 30
f (x) = x3 – 7x2 + 2x + 40
f (x) = x2 – 3x + 2
-1, 3, 4
-3, 1, 10
-2, 4, 5
1, 2

47. The Rational Zero Theorem
If a polynomial function has integer coefficients then every rational zero of the function has the following form:
P = factor of the constant term
Q factor of the leading coefficient

48. Find the rational zeros of
List the possible zeros
Test the zeros using synthetic division
Divide out the factor and factor the remaining trinomial to find the other zeros.
(You may use your calculator to guide you)

49. List all the possible rational zeros of the function.

50. Find all zeros of the function.

51. Molten Glass
At a factory, molten glass is poured into molds to make paperweights. Each mold is a rectangular prism whose height is 3 inches greater than the length of each side of the square base. A machine pours 20 cubic inches of liquid glass into each mold. What are the dimensions of the mold?

52. United States Exports
For 1980 through 1996, the total exports E (in billions of dollars) of the United States can be modeled by
where t is the number of years since In what year were the total exports about
$ billion?

53. Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n and n is greater than zero, then the equation f(x)=0 has at least one root in the set of complex numbers.
(written by Carl Friedrich Gauss)

54. When all real and imaginary solutions are counted (with all repeated solutions counted individually), an nth degree polynomial equation has exactly n solutions. Any nth degree polynomial function has exactly n zeros.

55. Turning points of a graph
The graph of every polynomial function of degree n has at most n-1 turning points. If the function has n distinct real zeros then its graph has exactly n-1 turning points.
Polynomial functions have local maximum and local minimum points, these are the turning points.
Quadratic functions have only one maximum or minimum point.

56. Finding Turning Points
Use your calculator to graph
Identify the x intercepts and the points where the local maximums and minimums occur.

57. Maximizing a Polynomial Model
You are designing an open box to be made of a piece of cardboard that is 10 inches by 15 inches. The box will be formed by making the cuts at the corners and folding up the sides so that the flaps are square. You want the box to have the greatest volume possible. How long should you make the cuts? What is the maximum volume? What will the dimensions of the finished box be?