properties of exponents
DESCRIPTION
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 . Examples. Polynomial Functions -exponents are whole numbers -coefficients are real numbers. - PowerPoint PPT PresentationTRANSCRIPT
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
nmnm aaa
mnnm aa )(
mmm baab )(
0,1
aa
am
m
0 1, 0a a
0, aaa
a nmn
m
0,
bb
a
b
am
mm
Examples
13
22
523
2
5
53
2
53
)(
)7(
)2()2(
5
3
)2(
yx
xy
bbb
s
r
Polynomial Functions-exponents are whole numbers-coefficients are real numbers
-2 is the leading coefficient4 is the degree ( the highest exponent)-7 is the constant term
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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
You can evaluate polynomial functions using-direct substitution
-synthetic substitution
• Evaluate 7582)( 24 xxxxf
EVALUATING A FUNCTIONgiven 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 – 72 + 15 -7
98
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
2 0 -8 5 -7
X = 3 2
The solution is the last number written.
66
1810
3035
105 98
• 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.3.525.00034.0000015.0 23 nnnt
11.3 seconds
End Behavior of Polynomial Functionsis 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
END BEHAVIORSWHAT 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)
Degreeeven
Leading Coefficient
positive End behavior of the function Graph of the function
Example: f (x) = x2
Degree even
Negative Leadingcoefficient
Example: f (x) = –x2
Degree Odd
Positive Leading coefficient
Example: f (x) = x3
Degree Odd
NegativeLeading coefficient
Example: f (x) = –x3
POLYNOMIAL GRAPHSIT’S A MATTER OF DEGREES
DEGREE/TYPE
0 /Constant y = 3 Horizontal 0 or infinity 0
END BEHAVIORSMAX # OF
ZEROSMAX
TURNING POINTSEXAMPLE
1/linear y = -2x + 4 Alternate 1 0
2/quadratic y = x2 + 2x – 1 Same 2 1
3/cubic y = x3 – 3x2 + 2 Alternate 3 2
4/quartic y = x4 – 4x3 – x2 + 12x – 2 Same 4 3
n (odd) Alternate n n - 1
n (even) Same n n - 1
6.3 OPERATIONS ON POLYNOMIALS
ADDITION: Aka: combine like terms
EXAMPLE: +
SUBTRACTION: add the opposite of the second polynomial
EXAMPLE: +
+
Horizontally: Vertically:
MULTIPLY
EXAMPLE: (
Horizontally: Vertically:
APPLICATIONSOF 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) = and E(t) = Where t is the number of years since 1985. Write a model for the net farm income, N, for those years
N(t) = G(t) - E(t)
N(t) = () - ()
N(t) =
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
Where t is the number of years since 1982. Write a model for the total revenue, R received from the sales of softbound books.
What was the total revenue from softbound books in 1990?
$7020 million ($7.02 billion)
𝑅 (𝑡)=.42704 𝑡3+5.44562 𝑡2+346.5166 𝑡+3679.92
Method #1: evaluate R with t = 8
Method #2: graph R and determine R(8)
R(t) = P(t) x N(t)
APPLICATION: BOOK BUSINESS
After vacation warm up
Simplify
Write in standard form
Graph
Use synthetic substitution to evaluate
for x=-2
8
52
3
)3(439
3
yx
xy
)44)(3( 2 xxx)342()7( 33 xxxx
13 xy
142 3 xxy
SPECIAL PRODUCT PATTERNSSUM 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 + 16NOTE: 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 + 15x² + 75x + 125
x³ + 3x²b(5) + 3x(25)² + 125
FACTORING REVIEW
COMMON FACTOR: 6x² + 15x + 27 = 3( )
TRINOMIAL: 2x² -5x – 12 = ( )( )
PERFECT SQUARE TRINOMIAL: x² + 20x + 100 = ( )( )
DIFFERENCE OF TWO SQUARES: x² - 49 = ( )( )
MORE SPECIAL FACTORING PATTERNS
SUM OF 2 CUBES: a³ + b³ = (a + b)(a² - ab + b²)
Example: x³ + 27 = (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 - 125x³ + 6427x³ - 8343x³ + 1000
Warm-up
• Factor
405
125
3108
27
3
3
2
3
x
x
xx
y
ZERO PRODUCT RULE(STILL GOOD!)
Solving polynomial equations:1. Transform equation to make one side zero2. Factor other side completely3. 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 = 0 x – 2 = 0 x + 2 = 0 x = 0 x = ±√3 x = 2 x = -2
𝑋 3+27=0(X + 3)(X² – 3X + 9) = 0
X + 3 = 0 OR X² – 3X + 9 = 0
FACTOR BY GROUPINGUse for polynomials with 4 terms
𝑟3−3𝑟 2+6𝑟 −18
Separate into 2 binomials: +
Factor out GCF of each: 𝑟2(𝑟 −3) + 6)
Factor out new GCF: (𝑟 −3)(𝑟2+6)
TRY THESE:
25
CHECK:(X² + 7)(X + 6)(z² - 16)(z – 2)(5p - 1)(5p + 1)(p – 1)(3m - 2)(3m + 2)(m + 2)
• 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.
• 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?
• 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?
• 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?
LONG DIVISION REVIEW
LONG DIVISION - Remember 4th grade?
32040 /15
Write dividend “inside the house”
Divide 1st digit(s) in dividend by the divisor; write answer in quotient
Multiply
Subtract
Bring down next digit
Repeat process as needed
3 2 0 4 015
3 0
2 0
5 41 5
4 59
Answer: 2136
2 1 3
0
6
9 0
0
POLYNOMIAL DIVISION
LONG DIVISION
(2x⁴ + 3x³ + 5x – 1) /(x² - 2x + 2)
Write dividend in standard form(include all degrees)Divide 1st term in dividend by 1st term in divisor
Multiply
Subtract
Bring down next term
Repeat process as needed
2x⁴ + 3x³ + 0x² + 5x – 1X² - 2x + 2
2x²
2x⁴ - 4x³ + 4x²
7x³ - 4x² +5x
+7x
10x² -9x -1
7x³ - 14x² + 14x
10x² - 20x + 20
11x - 21
+ 10
Answer: 2x² + 7x + 10 11x – 21X² - 2x + 2
• TRY THESE:Divide x² + 6x + 8 by x + 4
X + 6 R 6 or x + 6
Verify: (x – 3)(x + 6) + 6
X + 2
Verify: (x + 4)(x + 2)
Divide x² + 3x – 12 by x – 3
SOLUTIONS:
SYNTHETIC DIVISION:
Example: Divide x3 - 3x2 - 16x – 12 , by ( x – 6)
x3 - 3x2 - 16x – 12
1 -3 -16 -12
K = 6
1 6 3
18 2
12 0
Quotient: 1x2 + 3x + 2
- the other numbers in the answer are the coefficients/constant of the quotient
- The remainder is the last number written
- use the given value of k in the process below *- write only coefficients (including zeros)
- write polynomial expression in standard form (include all degree terms)
To divide polynomial f(x) by (x – k),
• TRY THESE:Divide x² + 6x + 8 by x + 4
X + 6 R 6 or x + 6
Verify: (x – 3)(x + 6) + 6
X + 2
Verify: (x + 4)(x + 2)
Divide x² + 3x – 12 by x – 3
SOLUTIONS:
RELATED THEOREMSREMAINDER 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)
1 2 -6 -9K = 2
1
2
4
8
2
4
-5 f(2) = -5
WORKOUT
2 6 -5 0 -60
-8 8 -12 48
-4
2 -2 3 -12 -12
Use the Remainder Theorem to evaluate P(-4) for P(x) = 2x4 + 6x3 – 5x2 - 60
P(-4) = -12
SPECIAL CASEUse the Remainder Theorem to evaluate P(-3) for P(x) = 2x3 + 11x2 + 18x + 9
2 11 18 9- 3
2-6 -15 -9 5 3 0
Quotient: 2x2 + 5x + 3
Note: the quotient is also factorable:2x2 + 5x + 3 = (2x + 3) (x + 1)
Since P(-3) = 0: 1. 3 is a zero of P(x) 2. (x - ¯3) is a factor (x + 3)
Therefore, 2x3 + 11x2 + 18x + 9 = (x + 3) (2x + 3) (x + 1)
Try: if one zero is 2
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OBSERVATIONWhen 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) b) (x3 – 4x2 + 3x + 2) (x + 2)
yes no
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
2 7 -33 -18
-6 -12 30 18
2 -5 -3 0
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)
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)
Warm-up
• Divide using long division.
)12()874(
)13()1523(
3
23
xxx
xxxx
• 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
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WRITING A FUNCTIONGIVEN 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
Try these:Given the zeros of a function, write the function.
1. -1, 3, 42. -3, 1, 103. -2, 4, 54. 1, 2
SOLUTIONS:1. f (x) = x3 - 6x2 + 5x + 122. f (x) = x3 – 8x2 – 23x + 303. f (x) = x3 – 7x2 + 2x + 40 4. f (x) = x2 – 3x + 2
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
• 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)•
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12,6,4,3,2,1
• List all the possible rational zeros of the function.
3 2
3 2
( ) 2 7 7 30
( ) 4 4
f x x x x
f x x x x
Find all zeros of the function.
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30114)(
23
23
xxxxf
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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?
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 1980. In what year were the total exports about
• $312.76 billion?
2332.23033.5131.0 23 tttE
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)
• 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.
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.
Finding Turning Points
• Use your calculator to graph
• Identify the x intercepts and the points where the local maximums and minimums occur.
23)( 23 xxxf
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?