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

7252)( 234 xxxxxf

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

1892)( 23 xxxxf

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

810112)( 23 xxxxf

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)•

12112)( 23 xxxxf

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.

5252)(

30114)(

23

23

xxxxf

xxxxf

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?