[PDF] - Properties of Binomial Theorem Exponents Review Factoring PDF Document

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Algebra II

Polynomials: Operations and Functions

www.njctl.org 2014-10-22

Slide 3 / 276 Table of Contents

Operations with Polynomials Review Dividing Polynomials Polynomial Functions Analyzing Graphs and Tables of Polynomial Functions Zeros and Roots of a Polynomial Function

click on the topic to go to that section

Special Binomial Products Properties of Exponents Review Writing Polynomials from its Given Zeros Binomial Theorem Factoring Polynomials Review

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Properties of Exponents Review

Return to Table of Contents This section is intended to be a brief review of this topic. For more detailed lessons and practice see Algebra 1.

Slide 5 / 276 Goals and Objectives

· Students will be able to simplify complex expressions containing exponents.

Slide 6 / 276 Why do we need this?

Exponents allow us to condense bigger expressions into smaller ones. Combining all properties of powers together, we can easily take a complicated expression and make it simpler.

SLIDE 2

Slide 7 / 276 Properties of Exponents

Product of Powers Power of Powers Power of a product Negative exponent Power of 0 Quotient of Powers

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1 Simplify: A 50m6q8 B 15m6q8 C 50m8q15 D Solution not shown . 5m2q3 10m4q5

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3 Divide: A B C D Solution not shown

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4 Simplify: A B C D Solution not shown

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Sometimes it is more appropriate to leave answers with positive exponents, and other times, it is better to leave answers without

fractions. You need to be able to translate expressions into

either form. Write with positive exponents: Write without a fraction:

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5 Simplify. The answer may be in either form. A B C D Solution not shown

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6 Simplify and write with positive exponents: A B C D Solution not shown

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When fractions are to a negative power, a short-cut is to invert the fraction and make the exponent positive. Try...

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Two more examples. Leave your answers with positive exponents.

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7 Simplify and write with positive exponents: A B C D Solution not shown

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Operations with Polynomials Review

Return to Table of Contents This section is intended to be a brief review of this topic. For more detailed lessons and practice see Algebra 1.

Slide 21 / 276 Goals and Objectives

· Students will be able to combine polynomial functions using operations of addition, subtraction, multiplication, and division.

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A monomial is an expression that is a number, a variable, or the product of a number and one or more variables with whole number exponents. A polynomial is the sum of one or more monomials, each of which is a term of the polynomial. Put a circle around each term:

Vocabulary Review Slide 23 / 276

Polynomials can be classified by the number of terms. The table below summarizes these classifications.

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Identify the degree of each polynomial:

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Polynomials can also be classified by degree. The table below summarizes these classifications.

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A polynomial function is a function in the form where n is a nonnegative integer and the coefficients are real numbers. The coefficient of the first term, an, is the leading coefficient. A polynomial function is in standard form when the terms are in

rder of degree from highest to lowest.

Polynomial Function Slide 28 / 276

Polynomial Functions Not Polynomial Functions Drag each relation to the correct box: f(x) = For extra practice, make up a few of your own!

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To add or subtract polynomials, simply distribute the + or - sign to each term in parentheses, and then combine the like terms from each polynomial. Examples: (2a

2 +3a - 9) + (a2 - 6a +3)

(2a2 +3a - 9) - (a

2 - 6a +3)

Watch your signs...forgetting to distribute the minus sign is one

f the most common mistakes students make !!

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Closure: A set is closed under an operation if when any two elements are combined with that operation, the result is also an element of the set. Is the set of all polynomials closed under

addition?
subtraction?

Explain or justify your answer.

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9 Simplify

A B C D

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12 What is the perimeter of the following figure? (answers are in units, assume all angles are right)

A B C D

2x - 3 8x2 - 3x + 4

10x + 1

x2 +5x - 2

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To multiply a polynomial by a monomial, you use the distributive property of multiplication over addition together with the laws of exponents. Example: Simplify.

2x(5x2 - 6x + 8)

(-2x)(5x2) + (-2x)(-6x) + (-2x)(8)

10x3 + 12x2 + -16x
10x3 + 12x2 - 16x

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13 What is the area of the rectangle shown?

A B C D

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14

A B C D

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15 Find the area of a triangle (A=

1/2bh) with a base of 5y

and a height of 2y + 2. All answers are in square units.

A B C D

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Compare multiplication of polynomials with multiplication of

integers. How are they alike and how are they different?

Is the set of polynomials closed under multiplication?

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=

Discuss how we could check this result.

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To multiply a polynomial by a polynomial, distribute each term of the first polynomial to each term of the second. Then, add like terms. Before combining like terms, how many terms will there be in each product below? 3 terms x 5 terms 5 terms x 8 terms 100 terms x 99 terms

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16 What is the total area of the rectangles shown? A B C D

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17

A B C D

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18

A B C D

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Example Part A: A town council plans to build a public parking lot. The outline below represents the proposed shape of the parking lot. Write an expression for the area, in square yards, of this proposed parking lot. Explain the reasoning you used to find the expression.

From High School CCSS Flip Book

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Example Part B: The town council has plans to double the area of the parking lot in a few years. They create two plans to do this. The first plan increases the length of the base of the parking lot by p yards, as shown in the diagram below. Write an expression in terms of x to represent the value of p, in feet. Explain the reasoning you used to find the value

f p.

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Example Part C: The town council’s second plan to double the area changes the shape of the parking lot to a rectangle, as shown in the diagram below. Can the value of z be represented as a polynomial with integer coefficients? Justify your reasoning.

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20 Find the value of the constant a such that

A 2 B 4 C 6 D -6

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Special Binomial Products

Return to Table of Contents

Slide 51 / 276 Square of a Sum

(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2 The square of a + b is the square of a plus twice the product of a and b plus the square of b. Example:

Slide 52 / 276 Square of a Difference

(a - b)2 = (a - b)(a - b) = a2 - 2ab + b2 The square of a - b is the square of a minus twice the product of a and b plus the square of b. Example:

Slide 53 / 276 Product of a Sum and a Difference

(a + b)(a - b) = a2 + -ab + ab + -b2 = Notice the sum of -ab and ab a2 - b2 equals 0. The product of a + b and a - b is the square of a minus the square of b. Example:

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+ = +2 +

2 2 2

Practice the square of a sum by putting any monomials in for and .

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=
2 +

2 2 2

Practice the square of a difference by putting any monomials in for and . How does this problem differ from the last? Study and memorize the patterns!! You will see them over and

ver again in many different ways.

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+ - =

2

2

Practice the product of a sum and a difference by putting any monomials in for and . How does this problem differ from the last two? This very important product is called the difference of squares.

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21

A B C D

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22 Simplify:

A B C D

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23 Simplify:

A B C D

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24 Multiply:

A B C D

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Challenge: See if you can work backwards to simplify the given problem without a calculator.

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Problem is from: Click for link for commentary and solution. A-APR Trina's Triangles

Alice and her friend Trina were having a conversation. Trina said "Pick any 2 integers. Find the sum of their squares, the difference of their squares and twice the product of the integers. These 3 numbers are the sides of a right triangle." Trina had tried this with several examples and it worked every time, but she wasn't sure this "trick" would always work.

a. Investigate Trina's conjecture for several pairs of integers. Does it

work?

b. If it works, then give a precise statement of the conjecture, using

variables to represent the chosen integers, and prove it. If not true, modify it so that it is true, and prove the new statement.

c. Use Trina's trick to find an example of a right triangle in which all of

the sides have integer length. all 3 sides are longer than 100 units, and the 3 side lengths do not have any common factors.

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Binomial Theorem

Return to Table of Contents

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The Binomial Theorem is a formula used to generate the expansion of a binomial raised to any power. Binomial Theorem Because the formula itself is very complex, we will see in the following slides some procedures we can use to simplify raising a binomial to any power.

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What happens when you multiply a binomial by itself n times? Evaluate:

n = 0 n = 3 n = 2 n = 1

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Let's try another one: Expand (x + y)4 What will be the exponents in each term of (x + y)5?

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25 The exponent of x is 5 on the third term of the expansion of . True False

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26 The exponents of y are decreasing in the expansion of True False

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27 What is the exponent of a in the fourth term of ?

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Pascal's Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 To get the next row, we start and end with 1, then add the two numbers above the next terms. Fill in the next 2 rows.... One way to find the coefficients when expanding a polynomial raised to the nth power is to use the nth row of Pascal's Triangle.

Row 0 Row 4

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28 All rows of Pascal's Triangle start and end with 1 True False

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29 What number is in the 5th spot of the 6th row of Pascal's Triangle?

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30 What number is in the 2nd spot of the 4th row of Pascal's Triangle?

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Now that we know how to find the exponents and the coefficients when expanding binomials, lets put it together. Expand

Teacher Notes

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Another Example Expand: (In this example, 2a is in place of x, and 3b is in place of y.)

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Now you try! Expand:

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31 What is the coefficient on the third term of the expansion

f

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33 The binomial theorem can be used to expand True False

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Factoring Polynomials Review

Return to Table of Contents

Slide 83 / 276 Factoring Polynomials Review

The process of factoring involves breaking a product down into its factors. Here is a summary of factoring strategies:

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Berry Method to factor Step 1: Calculate ac. Step 2: Find a pair of numbers m and n, whose product is ac, and whose sum is b. Step 3: Create the product . Step 4: From each binomial in step 3, factor out and discard any common factor. The result is your factored form. Example:

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Example: Step 1: ac = -15 and b = -2 Step 2: find m and n whose product is -15 and sum is -2; so m = -5 and n = 3 Step 3: (ax + m)(ax + n) = (3x - 5)(3x + 3) Step 4: (3x + 3) = 3(x + 1) so discard the 3 Therefore, 3x2 - 2x - 5 = (3x - 5)(x + 1)

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More factoring review.... (In this unit, sum or difference of cubes is not emphasized.)

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34 Factor out the GCF: 15m3n - 25m2 - 15mn3 A 15m(mn - 10m - n3) B 5m(3m2n - 5m - 3n3) C 5mn(3m2 - 5m - 3n2) D 5mn(3m2 - 5m - 3n) E 15mn(mn - 10m - n3)

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35 Factor: A (x - 5)(x - 5) B (x - 5)(x + 5) C (x + 15)(x + 10) D (x - 15)(x - 10) E Solution not shown x2 + 10x + 25

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36 Factor: A (n - 3)(m + 4n) B (n - 3)(m - 4n) C (n + 4)(m - n) D Not factorable E Solution not shown mn + 3m - 4n2 - 12n

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37 Factor: A (11m - 10n)(11m + 10m) B (121m - n)(m + 100n) C (11m - n)(11m + 100n) D Not factorable E Solution not shown 121m2 + 100n2

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38 Factor: A (11m - 10n)(11m + 10n) B (121m - n)(m + 100n) C (11m - n)(11m + 100n) D Not factorable E Solution not shown 121m2 - 100n2

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39 Factor: A (2x - 1)(5x - 3) B (2x + 1)(5x + 3) C (10x - 1)(x + 3) D (10x - 1)(x - 3) E Solution not shown 10x2 - 11x + 3

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40 Which expression is equivalent to 6x3 - 5x2y - 24xy2 + 20y3 ?

A x2 (6x - 5y) + 4y2 (6x + 5y) B x2 (6x - 5y) + 4y2 (6x - 5y) C x2 (6x - 5y) - 4y2 (6x + 5y) D x2 (6x - 5y) - 4y2 (6x - 5y)

From PARCC sample test

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41 Which expressions are factors of 6x3 - 5x2y - 24xy2 + 20y3 ? Select all that apply. A x2 + y2 B 6x - 5y C 6x + 5y D x - 2y E x + 2y

From PARCC sample test

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42 The expression x2(x - y)3 - y2(x - y)3 can be written in the form (x - y)a (x +y), where a is a

constant. What is the value of a?

From PARCC sample test

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Write the expression x - xy2 as the product of the greatest common factor and a binomial: Determine the complete factorization of x - xy2 :

From PARCC sample test

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Dividing Polynomials

Return to Table of Contents

Slide 98 / 276 Division of Polynomials

Here are 3 different ways to write the same quotient:

Slide 99 / 276 Slide 100 / 276 Examples

Click to Reveal Answer

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43 Simplify

A B C D

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44 Simplify

A B C D

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45 The set of polynomials is closed under division. True False

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46 Simplify.

A B C D

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47 Simplify.

A B C D

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52 If f (1) = 0 for the function, , what is the value of a?

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53 If f (3) = 27 for the function, , what is the value of a?

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Polynomial Functions

Return to Table of Contents

Slide 125 / 276 Goals and Objectives

· Students will be able to sketch the graphs of polynomial functions, find the zeros, and become familiar with the shapes and characteristics of their graphs.

Slide 126 / 276 Why We Need This

Polynomial functions are used to model a wide variety of real world phenomena. Finding the roots or zeros of a polynomial is one of algebra's most important problems, setting the stage for future math and science study.

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Slide 127 / 276 Graphs of Polynomial Functions

Features: · Continuous curve (or straight line) · Turns are rounded, not sharp Which are polynomials?

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The degree of a polynomial function and the coefficient

f the first term affect:

· the shape of the graph, · the number of turning points (points where the graph changes direction), · the end behavior, or direction of the graph as x approaches positive and negative infinity. If you have Geogebra on your computer, click below to go to an interactive webpage where you can explore graphs of polynomials.

The Shape of a Polynomial Function Slide 129 / 276 Slide 130 / 276 Optional Spreadsheet Activity

See the spreadsheet activity on the unit page for this unit entitled "Exploration of the values of the terms of a polynomial". Explore the impact of each term by changing values of the coefficients in row 1.

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Take a look at the graphs below. These are some of the simplest polynomial functions, y = xn. Notice that when n is even, the graphs are similar. What do you notice about these graphs? What would you predict the graph of y = x10 to look like? For discussion: despite appearances, how many points sit on the x-axis?

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Notice the shape of the graph y = xn when n is odd. What do you notice as n increases? What do you predict the graph of y = x21 would look like?

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Polynomials of Even Degree Polynomials of Odd Degree End behavior means what happens to the graph as x → and as x → - . What do you observe about end behavior?

∞ ∞ Slide 134 / 276

These are polynomials of even degree. Positive Lead Coefficient Negative Lead Coefficient Observations about end behavior?

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These are polynomials of odd degree. Observations about end behavior? Positive Lead Coefficient Negative Lead Coefficient

Slide 136 / 276 End Behavior of a Polynomial

Lead coefficient is positive Left End Right End Lead coefficient is negative Left End Right End Polynomial of even degree Polynomial of odd degree

Slide 137 / 276 End Behavior of a Polynomial

Degree: even Lead Coefficient: positive Degree: even Lead Coefficient: negative As x → ∞, f(x) → ∞ As x → -∞, f(x) → ∞ As x → ∞, f(x) → -∞ As x → -∞, f(x) → -∞ In other words, the function rises to the left and to the right. In other words, the function falls to the left and to the right.

Slide 138 / 276 End Behavior of a Polynomial

Degree: odd Lead Coefficient: positive Degree: odd Lead Coefficient: negative As As As As In other words, the function falls to the left and rises to the right. In other words, the function rises to the left and falls to the right.

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54 Determine if the graph represents a polynomial of odd

r even degree, and if the lead coefficient is

positive

r negative.

A

dd and

positive B

dd and

negative C even and positive D even and negative

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55 Determine if the graph represents a polynomial of

dd or even degree, and if the lead coefficient is

positive or negative.

A

dd and positive

B

dd and negative

C even and positive D even and negative

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56 Determine if the graph represents a polynomial of

dd or even degree, and if the lead coefficient is

positive or negative.

A

dd and positive

B

dd and negative

C even and positive D even and negative

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57 Determine if the graph represents a polynomial of

dd or even degree, and if the lead coefficient is

positive or negative.

A

dd and positive

B

dd and negative

C even and positive D even and negative

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Odd functions not only have the highest exponent that is odd, but all of the exponents are odd. An even function has only even exponents. Note: a constant has an even degree ( 7 = 7x

0)

Examples:

Odd function Even function Neither

f(x)=3x5 - 4x

3 + 2x h(x)=6x 4 - 2x 2 + 3 g(x)= 3x 2 + 4x - 4

y = 5x y = x2 y = 6x - 2 g(x)=7x

7 + 2x 3

f(x)=3x10 -7x

2

r(x)= 3x

5 +4x 3 -2

Odd and Even Functions Slide 144 / 276

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An even function is symmetric about the y-axis. Definition of an Even Function

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60 Choose all that apply to describe the graph.

A

Odd Degree

B

Odd Function

C

Even Degree

D

Even Function

E

Positive Lead Coefficient

F

Negative Lead Coefficient

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61 Choose all that apply to describe the graph. A Odd Degree B Odd Function C Even Degree D Even Function E Positive Lead Coefficient F Negative Lead Coefficient

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62 Choose all that apply to describe the graph.

A

Odd Degree

B

Odd Function

C

Even Degree

D

Even Function

E

Positive Lead Coefficient

F

Negative Lead Coefficient

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63 Choose all that apply to describe the graph. A Odd Degree B Odd Function

C

Even Degree

D

Even Function

E

Positive Lead Coefficient F Negative Lead Coefficient

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64 Choose all that apply to describe the graph. A Odd Degree B Odd Function

C

Even Degree

D

Even Function

E

Positive Lead Coefficient F Negative Lead Coefficient

Slide 153 / 276 Zeros of a Polynomial

"Zeros" are the points at which the polynomial intersects the x-axis. They are called "zeros" because at each point f (x) = 0. Another name for a zero is a root. A polynomial function of degree n has at MOST n real zeros. An odd degree polynomial must have at least one real zero. (WHY?) Zeros

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A polynomial function of degree n has at MOST n - 1 turning points, also called relative maxima and relative minima. These are points where the graph changes from increasing to decreasing, or from decreasing to increasing.

Relative Maxima and Minima

Relative Maxima Relative Minima

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65 How many zeros does the polynomial appear to have?

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66 How many turning points does the polynomial appear to have?

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67 How many zeros does the polynomial appear to have?

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68 How many turning points does the graph appear to have? How many of those are relative minima?

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69 How many zeros does the polynomial appear to have?

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70 How many turning points does the polynomial appear to have? How many of those are relative maxima?

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71 How many zeros does the polynomial appear to have?

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72 How many relative maxima does the graph appear to have? How many relative minima?

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Analyzing Graphs and Tables

f Polynomial Functions

Return to Table of Contents

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x y

3

58

2

19

1
5

1

2

2 3 3 4 4

5

A polynomial function can be sketched by creating a table, plotting the points, and then connecting the points with a smooth curve. Look at the first term to determine the end behavior of the graph. In this case, the coefficient is negative and the degree is odd, so the function rises to the left and falls to the right.

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x y

3

58

2

19

1
5

1

2

2 3 3 4 4

5

How many zeros does this function appear to have?

Answer

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x y

3

58

2

19

1
5

1

2

2 3 3 4 4

5

There is a zero at x = -1, a second between x = 1 and x = 2, and a third between x = 3 and x = 4. How can we recognize zeros given only a table? Answer

Slide 167 / 276 Intermediate Value Theorem

Given a continuous function f(x), every value between f(a) and f(b) exists. Let a = 2 and b = 4, then f(a)= -2 and f(b)= 4. For every x-value between 2 and 4 there exists a y-value, so there must be an x-value for which y = 0.

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x y

3

58

2

19

1
5

1

2

2 3 3 4 4

5

The Intermediate Value Theorem justifies the statement that there is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4.

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73 How many zeros of the continuous polynomial given can be found using the table? x y

3
12
2
4
1

1 3 1 2

2

3 4 4

5

Answer

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74 If the table represents a continuous function, between which two values of x can you find the smallest x-value at which a zero occurs? A

3

B

2

C

1

D E

1

F

2

G

3

H

4 x y

3
12
2
4
1

1 3 1 2

2

3 4 4

5

Answer

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75 How many zeros of the continuous polynomial given can be found using the table? x y

3

2

2
1

5 2 1

3

2 4 3 4 4

5

Answer

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76 According to the table, what is the least value of x at which a zero occurs on this continuous function?

A

3

B

2

C

1

D E

1

F

2

G

3

H

4 x y

3

2

2
1

5 2 1

3

2 4 3 4 4

5

Answer

Slide 173 / 276 Relative Maxima and Relative Minima

There are 2 relative maximum points at x = -1 and at x = 1. The relative maximum value appears to be -1 (the y-coordinate). There is a relative minimum at (0, -2).

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How do we recognize the relative ma xima and minima from a table?

x f(x)

3

5

2

1

1
1
4

1

5

2

2

3 2 4

In the table, as x goes from -3 to 1, f(x) is

decreasing. As

x goes from 1 to 3, f(x) is

increasing. And a

s x goes from 3 to 4, f(x) is decreasing. The relative maxima and minima

ccur when the direction changes

from decreasing to increasing, or from increasing to decreasing. The y-coordinate indicates this change in direction as its value rises

r falls.

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77 At approximately what x-values does a relative minimum occur? A

3

B

2

C

1

D E

1

F

2

G

3

H

4

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78 At about what x-values does a relative maximum

ccur?

A

3

B

2

C

1

D E

1

F

2

G

3

H

4

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79 At about what x-values does a relative minimum

ccur?

A

3

B

2

C

1

D E

1

F

2

G

3

H

4 x y

3

5

2

1

1
1
4

1

5

2

2

3 2 4

Answer

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80 At about what x-values does a relative maximum

ccur?

A

3

B

2

C

1

D E

1

F

2

G

3

H

4 x y

3

5

2

1

1
1
4

1

5

2

2

3 2 4

Answer

SLIDE 31

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81 At about what x-values does a relative minimum

ccur?

A

3

B

2

C

1

D E 1 F 2 G 3 H 4 x y

3

2

2
1

5 2 1

3

2 4 3 4 4

5

Answer

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82 At about what x-values does a relative maximum

ccur?

A

3

B

2

C

1

D E

1

F

2

G

3

H

4 x y

3

2

2
1

5 2 1

3

2 4 3 5 4

5

Answer

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Zeros and Roots of a Polynomial Function

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Slide 184 / 276 Real Zeros of Polynomial Functions

For a function f(x) and a real number a, if f (a) = 0, the following statements are equivalent: x = a is a zero of the function f(x). x = a is a solution of the equation f (x) = 0. (x - a) is a factor of the function f(x). (a, 0) is an x-intercept of the graph of f(x).

Slide 185 / 276 The Fundamental Theorem of Algebra

An imaginary zero occurs when the solution to f (x) = 0 contains complex numbers. Imaginary zeros are not seen on the graph. If f (x) is a polynomial of degree n, where n > 0, then f (x) = 0 has n zeros including multiples and imaginary zeros.

Slide 186 / 276 Complex Numbers

Complex numbers will be studied in detail in the Radicals Unit. But in order to fully understand polynomial functions, we need to know a little bit about complex numbers. Up until now, we have learned that there is no real number, x, such that x2 = -1. However, there is such a number, known as the imaginary unit, i, which satisfies this equation and is defined as . The set of complex numbers is the set of numbers of the form a + bi , where a and b are real numbers. When a = 0, bi is called a pure imaginary number.

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The square root of any negative number is a complex number. For example, find a solution for x2 = -9:

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Complex Numbers Real Imaginary 3i 2 - 4i

0.765

2/3

11

9+6i Drag each number to the correct place in the diagram.

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The number of the zeros of a polynomial, both real and imaginary, is equal to the degree of the polynomial. This is the graph of a polynomial with degree 4. It has four unique zeros: -2.25, -.75, .75, 2.25 Since there are 4 real zeros, there are no imaginary zeros. (4 in total - 4 real = 0 imaginary)

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This 5th degree polynomial has 5 zeros, but only 3 of them are real. Therefore, there must be two imaginary. (How do we know that this is a 5th degree polynomial?) Note: imaginary roots always come in pairs: if a + bi is a root, then a - bi is also a root. (These are called conjugates - more

n that in later units.)

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This is a 4th-degree

polynomial. It has two

unique real zeros: -2 and 2. These two zeros are said to have a multiplicity of two, which means they each occur twice. There are 4 real zeros and therefore no imaginary zeros for this function.

2 zeros each

A vertex on the x-axis indicates a multiple zero, meaning the zero

ccurs

two or more times.

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What do you think are the zeros and their multiplicity for this function?

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Notice the function for this graph. x - 1 is a factor two times, and x = 1 is a zero twice. x + 2 is a factor two times, and x = -2 is a zero twice. Therefore, 1 and -2 are zeros with multiplicity of 2. x + 3 is a factor once, and x = 3 is a zero with multiplicity of 1.

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83 How many real zeros does the 4th-degree polynomial graphed have?

A B

1

C

2

D

3

E

4

F

5

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84 Do any of the zeros have a multiplicity of 2 ?

Yes No

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85 How many imaginary zeros does this 7th degree polynomial have?

A B

1

C

2

D

3

E

4

F

5

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86 How many real zeros does the 3rd degree polynomial have?

A B

1

C

2

D

3

E

4

F

5

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87 Do any of the zeros have a multiplicity of 2 ?

Yes No

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88 How many imaginary zeros does the 5th degree polynomial have?

A B

1

C

2

D

3

E

4

F

5

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89 How many imaginary zeros does this 4

th-degree

polynomial have?

A B

1

C

2

D

3

E

4

F

5

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90 How many real zeros does the 6th degree polynomial have?

A B

1

C

2

D

3

E

4

F

6

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91 Do any of the zeros have a multiplicity of 2 ?

Yes No

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92 How many imaginary zeros does the 6th degree polynomial have?

A B

1

C

2

D

3

E

4

F

5

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Recall the Zero Product Property. If the product of two or more quantities or factors equals 0, then at least one of the quantities must equal 0.

Finding the Zeros from an Equation in Factored Form:

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So, if , then the zeros of are 0 and -1. So, if , then the zeros of are ___ and ___.

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Find the zeros, including multiplicities, of the following polynomial.

r
r
r
r

Don't forget the ±!!

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Slide 211 / 276 Slide 212 / 276 Slide 213 / 276 Slide 214 / 276 Slide 215 / 276 Slide 216 / 276

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Slide 217 / 276 Slide 218 / 276 Slide 219 / 276 Slide 220 / 276 Slide 221 / 276

Find the zeros, showing the multiplicities, of the following polynomial.

r
r
r
r

This polynomial has two distinct real zeros: 0 and 1. This is a 3rd degree polynomial, so there are 3 zeros (count 1 twice). 1 has a multiplicity of 2. 0 has a multiplicity of 1. There are no imaginary zeros. To find the zeros, you must first write the polynomial in factored form.

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Find the zeros, including multiplicities, of the following polynomial.

r
r
r

There are two distinct real zeros: , both with a multiplicity of 1. There are two imaginary zeros: , both with a multiplicity of 1. This polynomial has 4 zeros.

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105 How many zeros does the polynomial function have? A 0 B 1 C 2 D 3 E 4

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106 How many REAL zeros does the polynomial equation have? A 0 B 1 C 2 D 3 E 4

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107 What are the zeros and their multiplicities of the polynomial function ? A x = -2, mulitplicity of 1 B x = -2, multiplicity of 2 C x = 3, multiplicity of 1 D x = 3, multiplicity of 2 E x = 0, multiplicity of 1 F x = 0, multiplicity of 2

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108 Find the solutions of the following polynomial equation, including multiplicities. A x = 0, multiplicity of 1 B x = 3, multiplicity of 1 C x = 0, multiplicity of 2 D x = 3, multiplicity of 2

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109 Find the zeros of the polynomial equation, including multiplicities: A x = 2, multiplicity 1 B x = 2, multiplicity 2 C x = -i, multiplicity 1 D x = i, multiplicity 1 E x = -i, multiplcity 2 F x = i, multiplicity 2

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110 Find the zeros of the polynomial equation, including multiplicities: A 2, multiplicity of 1 B 2, multiplicity of 2 C -2, multiplicity of 1 D -2, multiplicity of 2 E , multiplicity of 1 F - , multiplicity of 1

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Find the zeros, showing the multiplicities, of the following polynomial. To find the zeros, you must first write the polynomial in factored form. However, this polynomial cannot be factored using normal methods. What do you do when you are STUCK??

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We are going to need to do some long division, but by what do we divide? The Remainder Theorem told us that for a function, f (x), if we divide f (x) by x - a, then the remainder is f (a). If the remainder is 0, then x - a if a factor of f (x). In other words, if f (a) = 0, then x - a is a factor of f (x). So how do we figure out what a should be???? We could use guess and check, but how can we narrow down the choices?

Slide 231 / 276 The Rational Zeros Theorem:

Let with integer coefficients. There is a limited number of possible roots or zeros. · Integer zeros must be factors of the constant term, a0. · Rational zeros can be found by writing and simplifying fractions where the numerator is an integer factor of a0 and the denominator is an integer fraction of an.

Slide 232 / 276 RATIONAL ZEROS THEOREM

Make list of POTENTIAL rational zeros and test them out. Potential List: Hint: To check for zeros, first try the smaller integers -- they are easier to work with.

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therefore (x -1) is a factor of the polynomial. Use POLYNOMIAL DIVISION to factor out. Using the Remainder Theorem, we find that 1 is a zero:

r
r
r
r

This polynomial has three distinct real zeros: -2, -1/3, and 1, each with a multiplicity of 1. There are no imaginary zeros.

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Find the zeros using the Rational Zeros Theorem, showing the multiplicities, of the following polynomial. Potential List:

± ±1

3 is a distinct zero, therefore (x + 3) is a factor.

Use POLYNOMIAL DIVISION to factor out. Hint: since all of the signs in the polynomial are +, only negative numbers will work. Try -3:

SLIDE 40

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r
r
r
r

This polynomial has two distinct real zeros: -3, and -1.

3 has a multiplicity of 2 (there are 2 factors of x + 3).
1 has a multiplicity of 1.

There are no imaginary zeros.

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111 Which of the following is a zero of A x = -1 B x = 1 C x = 7 D x = -7

?

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112 Find the zeros of the polynomial equation, including multiplicities, using the Rational Zeros Theorem A x = 1, multiplicity 1 B x = 1, mulitplicity 2 C x = 1, multiplicity 3 D x = -3, multiplicity 1 E x = -3, multiplicity 2 F x = -3, multiplicity 3

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113 Find the zeros of the polynomial equation, including multiplicities, using the Rational Zeros Theorem A x = -2, multiplicity 1 B x = -2, multiplicity 2 C x = -2, multiplicity 3 D x = -1, multiplicity 1 E x = -1, multiplicity 2 F x = -1, multiplicity 3 Pull for Ans wer

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114 Find the zeros of the polynomial equation, including multiplicities, using the Rational Zeros Theorem A x = 1, multiplicity 1 B x = -1, multiplicity 1 C x = 3, multiplicity 1 D x = -3, multiplicity 1 E x = , multiplicity 1 F x = , multiplicity 1 G x = , multiplicity 1 H x = , multiplicity 1

Pull for Answer

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116 Find the zeros of the polynomial equation.

A x = 2 B x = -2 C x =3 D x = -3 E x = 3i F x = -3i G x = H x = -

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Writing a Polynomial Function from its Given Zeros

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Slide 243 / 276 Goals and Objectives

· Students will be able to write a polynomial from its given zeros.

Slide 244 / 276

Write (in factored form) the polynomial function of lowest degree using the given zeros, including any multiplicities. x = -1, multiplicity of 1 x = -2, multiplicity of 2 x = 4, multiplicity of 1

r
r
r
r
r
r

Work backwards from the zeros to the original polynomial. For each zero, write the corresponding factor.

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117 Write the polynomial function of lowest degree using the zeros given.

A B C D x = -.5, multiplicity of 1 x = 3, multiplicity of 1 x = 2.5, multiplicity of 1

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118 Write the polynomial function of lowest degree using the zeros given. A B C D

x = 1/3, multiplicity of 1 x = -2, multiplicity of 1 x = 2, multiplicity of 1

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119 Write the polynomial function of lowest degree using the zeros given. A B C D E

x = 0, multiplicity of 3 x = -2, multiplicity of 2 x = 2, multiplicity of 1 x = 1, multiplicity of 1 x = -1, multiplicity of 2

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Write the polynomial function of lowest degree using the zeros from the given graph, including any multiplicities.

x = -2

x = -1 x = 1.5 x = 3

x = -2 x = -1 x = 1.5 x = 3

r
r
r

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120 Write the polynomial function of lowest degree using the zeros from the given graph, including any multiplicities. A B C D E F

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121 Write the polynomial function of lowest degree using the zeros from the given graph, including any multiplicities. A B C D

Slide 251 / 276 Slide 252 / 276 Match each graph to its equation.

y = x2 + 2 y = (x + 2)2 y = (x-1)(x-2)(x-3) y = (x-1)(x-2)(x-3)2

SLIDE 43

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Sketch the graph of f(x) = (x-1)(x-2)2. After sketching, click on the graph to see how accurate your sketch is.

Sketch Slide 254 / 276

Analyzing Graphs using a Graphing Calculator Enter the function into the calculator (Hit y= then type). Check your graph, then set the window so that you can see the zeros and the relative minima and maxima. (Look at the table to see what the min and max values of x and y should be.) Use the Calc functions ( 2nd TRACE ) to find zeros: Select 2: Zero Your graph should appear. The question "Left Bound?" should be at the bottom of the screen. Use the left arrow to move the blinking cursor to the left side of the zero and press ENTER. The question "Right Bound?" should be at the bottom of the screen. Use the right arrow to move the blinking cursor to the right side of the zero and press ENTER. The question "Guess?" should be at the bottom of the screen. Press ENTER again, and the coordinates of the zero will be given.

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Use the Calc functions (2nd TRACE) to find relative min or max: Select 3: minimum or 4: maximum. Your graph should appear. The question "Left Bound?" should be at the bottom of the screen. Use the left arrow to move the blinking cursor to the left side of the turning point and press ENTER. The question "Right Bound?" should be at the bottom of the screen. Use the right arrow to move the blinking cursor to the right side of the turning point and press ENTER. The question "Guess?" should be at the bottom of the screen. Press ENTER again, and the coordinates of the min or max will be given. Finding Minima and Maxima

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Use a graphing calculator to find the zeros and turning points of Note: The calculator will give an estimate. Rounding may be needed.

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Use a graphing calculator to find the zeros and turning points of

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Sketch the graph of f(x) = (x-1)(x+1)(x-2)(x+2)(x-3)(x+3)(x-4). After sketching, click on the graph to see how accurate your sketch is.

SLIDE 44

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The product of 4 positive consecutive integers is 175,560. Write a polynomial equation to represent this problem. Use a graphing utility or graphing calculator to find the numbers. Hint: set your equation equal to zero, and then enter this equation into the calculator. How could you use a calculator and guess and check to find the answer to this problem?

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An open box is to be made from a square piece of cardboard that measures 50 inches on a side by cutting congruent squares of side- length x from each corner and folding the sides.

50

x

1. Write the equation of a polynomial

function to represent the volume of the completed box.

2. Use a graphing calculator or graphing

utility to create a table of values for the height of the box. (Consider what the domain of x would be.) Use the table to determine what height will yield the maximum volume.

3. Look at the graph and calculate the

maximum volume within the defined

domain. Does this answer match your

answer above? (Use the table values to determine how to set the viewing window.)

x

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An engineer came up with the following equation to represent the height, h(x), of a roller coaster during the first 300 yards of the ride: h(x) = -3x4 + 21x3 - 48x2 + 36x, where x represents the horizontal distance of the roller coaster from its starting place, measured in 100's of yards. Using a graphing calculator or a graphing utility, graph the function on the interval 0 < x < 3. Sketch the graph below. Does this roller coaster look like it would be fun? Why or why not?

Derived from

( (

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For what values of x is the roller coaster 0 yards off the ground? What do these values represent in terms of distance from the beginning of the ride? Verify your answers above by factoring the polynomial h(x) = -3x4 + 21x3 - 48x2 + 36x

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How do you think the engineer came up with this model? Why did we restrict the domain of the polynomial to the interval from 0 to 3? In the real world, what is wrong with this model at a distance of 0 yards and at 300 yards?

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Consider the function f(x) = x3 - 13x2 + 44x - 32. Use the fact that x - 4 is a factor to factor the polynomial. What are the x-intercepts for the graph of f ? At which x-values does the function change from increasing to decreasing and from decreasing to increasing?

SLIDE 45

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How can we tell if a function is positive or negative on an interval between x-intercepts? Given our polynomial f(x) = x3 - 13x2 + 44x - 32... When x < 1, is the graph above or below the x-axis? When 1 < x < 4, is the graph above or below the x-axis? When 4 < x < 8, is the graph above or below the x-axis? When x > 8, is the graph above or below the x-axis?

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123 Consider the function f (x)=(2x -1)(x + 4)(x - 2). What is the y-intercept of the graph of the function in the coordinate plane?

From PARCC sample test

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Consider the function f (x)=(2x -1)(x + 4)(x - 2). For what values of x is f (x) >0? Use the line segments and endpoint indicators to build the number line that answers the question.

From PARCC sample test

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124 Consider the function f (x)=(2x -1)(x + 4)(x - 2). What is the end behavior of the graph of the function?

A B C D

From PARCC sample test

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125 Consider the function f (x)=(2x -1)(x + 4)(x - 2). How many relative maximums does the function have?

A none B one C two D three

From PARCC sample test

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How many relative maxima and minima? f(x) = (x+1)(x-3) g(x) = (x-1)(x+3)(x-4) h(x) =x (x-2)(x-5)(x+4)

Degree: # x-intercepts: # turning points:

Observations: f(x) g(x) h(x) Answer

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Degree: # x-intercepts: # turning points:

Observations:

f(x) g(x) h(x)

How many relative maxima and minima? Answer

Slide 272 / 276 Increasing and Decreasing

Given a function f whose domain and range are subsets of the real numbers and I is an interval contained within the domain, the function is called increasing on the interval if f (x1) < f (x2) whenever x1 < x2 in I. It is called decreasing on the interval if f (x1) > f (x2) whenever x1 < x2 in I.

Restate this in your own words:

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Mark on this graph and state using inequality notation the intervals that are increasing and those that are decreasing.

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126 Select all of the statements that are true based on the graph provided:

A The degree of the function is

even.

B There are 4 turning points. C The function is increasing

n the interval from x = -1

to x = 2.4.

D The function is increasing

when x < -1.

E x - 2 and x + 3 are factors of the polynomial

that defines this function.

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127 Given the function f (x) = x7 - 4x5 - x3 + 4x. Which of the following statements are true? Select all that apply.

A As x → ∞, f (x) → ∞. B There are a maximum of 6 real zeros for this

function.

C x = -1 is a solution to the equation f (x) = 0. D The maximum number of relative minima and

maxima for this function is 7.

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For each function described by the equations and graphs shown, indicate whether the function is even, odd, or neither even nor odd: k(x)

h(x)

f(x)=3x2

g(x)=-x 3 + 5

Even Odd Neither

f(x)

g(x)

h(x) k(x) k(x)

Answer

From PARCC sample test