Algebra II Polynomials: Operations and Functions 2014-10-22 - - PDF document

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Algebra II

Polynomials: Operations and Functions

www.njctl.org 2014-10-22

Slide 2 / 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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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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Teacher Notes IMPORTANT TIP: Throughout this unit, it is extremely important that you, as a teacher, emphasize correct vocabulary and make sure students truly know the difference between monomials and polynomials. Having a solid understanding of rules that accompany each will give them a strong foundation for future math classes.

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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 4 / 276 Goals and Objectives

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

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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 6 / 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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1 Simplify: A 50m6q8 B 15m6q8 C 50m8q15 D Solution not shown . 5m2q3 10m4q5

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Answer

A Slide 9 (Answer) / 276 Slide 10 / 276

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3 Divide: A B C D Solution not shown

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3 Divide: A B C D Solution not shown

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Answer

C Slide 11 (Answer) / 276

4 Simplify: A B C D Solution not shown

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4 Simplify: A B C D Solution not shown

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Answer

D Slide 12 (Answer) / 276

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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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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Answer

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

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

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Answer

C Slide 14 (Answer) / 276

6 Simplify and write with positive exponents: 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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Answer

D Slide 15 (Answer) / 276

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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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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Answer

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Two more examples. Leave your answers with positive exponents.

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Two more examples. Leave your answers with positive exponents.

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Answer

= = Slide 17 (Answer) / 276

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

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

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Answer

B Slide 18 (Answer) / 276 Slide 19 / 276

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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 20 / 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:

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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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Identify the degree of each polynomial:

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Answer

3+2=5th degree 5+1=6th degree

Not a polynomial

4th degree

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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.

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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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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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Answer

Yes to both. When you add

• r subtract polynomials,

the answer will always be a polynomial. Discuss with examples, and try to find a counterexample.

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9 Simplify

A B C D

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9 Simplify

A B C D

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Answer A

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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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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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Answer

D Slide 34 (Answer) / 276

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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13 What is the area of the rectangle shown?

A B C D

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Answer

A Slide 36 (Answer) / 276

14

A B C D

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14

A B C D

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Answer

A

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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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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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Answer

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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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? Teacher Notes

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Notice how the distributive property is used in both

• examples. Each term in the

factor (2x + 2) must be multiplied by each term in the factor (2x2 + 4x + 3), just like the value of each digit of 22 must be multiplied by each digit of 243.

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=

Discuss how we could check this result.

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=

Discuss how we could check this result.

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Teacher Notes Encourage students to substitute a value for x in each expression,

• btaining the same 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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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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Answer 15, 40, 9900

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

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

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Answer

D

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17

A B C D

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17

A B C D

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Answer

B

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18

A B C D

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18

A B C D

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Answer

C

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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 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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Answer

Sample Response: Part A Missing vertical dimension is 2x # 5 # (x # 5) = x. Area = x(x # 5) + x(2x + 15) = x

2 # 5x + 2x 2 + 15x

= 3x

2 + 10x square yards

*Let students work on Parts A,B and C in their groups and share their work with the

• class. This problem should take more time

than typical slides. Give students the time they need to complete the problems.

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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 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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Answer

Part B Doubled area = 6x

2 + 20x square yards.

Area of top left corner = x

2 # 5x square yards.

Area of lower portion with doubled area = 6x

2 + 20x #

(x

2 # 5x)

= 5x

2 + 25x square yards

Since the width remains x yards, the longest length must be (5x

2 + 25x) ÷ x = 5x + 25 yards long.

So, y = 5x + 25 # (2x + 15) = 5x + 25 # 2x # 15 = 3x + 10 yards.

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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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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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Answer

Part C If z is a polynomial with integer coefficients, the length of the rectangle, 2 x + 15 + z, would be a factor of the doubled area. Likewise, 2 x # 5 would be a factor of the doubled area. But 2x # 5 is not a factor of 6 x

x. So 2x + 15 + z is not a factor either. Therefore, z cannot be represented as a polynomial with integer coefficients.

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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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20 Find the value of the constant a such that

A 2 B 4 C 6 D -6

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Answer

A

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Special Binomial Products

Return to Table of Contents

Slide 50 / 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 51 / 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:

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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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21

A B C D

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Answer

B

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22 Simplify:

A B C D

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22 Simplify:

A B C D

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Answer

D

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23 Simplify:

A B C D

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23 Simplify:

A B C D

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Answer

C

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24 Multiply:

A B C D

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24 Multiply:

A B C D

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Answer

A

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

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

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Answer Rewrite as

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

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

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Answer

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

Answer

True

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

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

Answer

False

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

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

Answer

7

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

Answer

True

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

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

Answer

15

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30 What number is in the 2nd spot of the 4th 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?

Answer

4 Slide 75 (Answer) / 276

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

Since the exponent is 5, we are going to use the fifth row of Pascal's triangle as the coefficients. Combining this with the increasing and decreasing exponents, we get:

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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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33 The binomial theorem can be used to expand True False

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Answer

False

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Factoring Polynomials Review

Return to Table of Contents

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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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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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Answer

B

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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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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

Answer

E

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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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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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Answer

E mn + 3m - 4n2 - 12n = m(n + 3) - 4n(n + 3) = (n + 3)(m - 4n)

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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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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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Answer

D Not factorable because it is a sum of squares with no GCF

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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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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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Answer

A

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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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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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Answer ac = 30, b = -11 so m = -5, n = -6 (10x - 5)(10x - 6) =(2x - 1)(5x - 3)

A

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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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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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Answer

D

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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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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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Answer

B, D, E

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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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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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Answer

4 Slide 95 (Answer) / 276

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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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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Answer

x(1 - y2) x(1 - y)(1 + y) Slide 96 (Answer) / 276

Dividing Polynomials

Return to Table of Contents

Slide 97 / 276 Division of Polynomials

Here are 3 different ways to write the same quotient:

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Examples

Click to Reveal Answer

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43 Simplify

A B C D

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43 Simplify

A B C D

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Answer

D

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44 Simplify

A B C D

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44 Simplify

A B C D

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Answer

A

Slide 102 (Answer) / 276

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

Slide 103 / 276

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

[This object is a pull tab]

Answer

False

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Slide 106 / 276 Slide 107 / 276 Slide 108 / 276

Slide 109 / 276 Slide 110 / 276 Slide 111 / 276

Slide 111 (Answer) / 276 Slide 112 / 276 Slide 112 (Answer) / 276

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46 Simplify.

A B C D

Slide 116 / 276

46 Simplify.

A B C D

[This object is a pull tab]

Answer

B

Slide 116 (Answer) / 276

47 Simplify.

A B C D

Slide 117 / 276

47 Simplify.

A B C D

[This object is a pull tab]

Answer

B

Slide 117 (Answer) / 276 Slide 118 / 276 Slide 118 (Answer) / 276

Slide 119 / 276 Slide 119 (Answer) / 276 Slide 120 / 276

Slide 120 (Answer) / 276 Slide 121 / 276 Slide 121 (Answer) / 276

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

Slide 122 / 276

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

[This object is a pull tab]

Answer

substitute 1 for x and solve for a. a = 0.

Slide 122 (Answer) / 276

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

Slide 123 / 276

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

[This object is a pull tab]

Answer

a = 1

Slide 123 (Answer) / 276

Polynomial Functions

Return to Table of Contents

Slide 124 / 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 125 / 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.

Slide 126 / 276 Graphs of Polynomial Functions

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

Slide 127 / 276

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 128 / 276

Slide 129 / 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.

Slide 130 / 276

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?

Slide 131 / 276

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?

[This object is a pull tab]

Answer

Slide 131 (Answer) / 276

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?

Slide 132 / 276

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?

[This object is a pull tab]

Answer

Slide 132 (Answer) / 276

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 133 / 276

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?

∞ ∞

[This object is a pull tab]

Answer

Even degree - both ends are going in the same direction (both up or both down). Odd degree - one end is up and the other down.

Slide 133 (Answer) / 276

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

Slide 134 / 276

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

Slide 135 / 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 136 / 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 137 / 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.

Slide 138 / 276

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

Slide 139 / 276

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

[This object is a pull tab]

Answer

D even and negative

Slide 139 (Answer) / 276

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

Slide 140 / 276

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

[This object is a pull tab]

Answer

A odd and positive

Slide 140 (Answer) / 276

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

Slide 141 / 276

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

[This object is a pull tab]

Answer

C even and positive

Slide 141 (Answer) / 276

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

Slide 142 / 276

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

[This object is a pull tab]

Answer

B

• dd and negative

Slide 142 (Answer) / 276

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 143 / 276 Slide 144 / 276 Slide 144 (Answer) / 276

Slide 145 / 276 Slide 145 (Answer) / 276 Slide 146 / 276

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

Slide 147 / 276

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

Slide 148 / 276

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

[This object is a pull tab]

Answer

A Odd- Degree B Odd- Function E Positive Lead Coefficient

Slide 148 (Answer) / 276

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

Slide 149 / 276

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

[This object is a pull tab]

Answer

C Even- Degree D Even- Function E Positive Lead Coefficient

Slide 149 (Answer) / 276

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

Slide 150 / 276

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

[This object is a pull tab]

Answer

A Odd- Degree B Odd- Function F Negative Lead Coefficient

Slide 150 (Answer) / 276

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

Slide 151 / 276

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

[This object is a pull tab]

Answer

A Odd- Degree E Positive Lead Coefficient

Slide 151 (Answer) / 276

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 152 / 276

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

[This object is a pull tab]

Answer

C Even- Degree D Even - Function F Negative Lead Coefficient

Slide 152 (Answer) / 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

Slide 153 / 276

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

Slide 154 / 276

65 How many zeros does the polynomial appear to have?

Slide 155 / 276

65 How many zeros does the polynomial appear to have?

[This object is a pull tab]

Answer

5 zeros

Slide 155 (Answer) / 276

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

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

[This object is a pull tab]

Answer

4 turning points

Slide 156 (Answer) / 276

67 How many zeros does the polynomial appear to have?

Slide 157 / 276

67 How many zeros does the polynomial appear to have?

[This object is a pull tab]

Answer

4 zeros

Slide 157 (Answer) / 276

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

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

[This object is a pull tab]

Answer

3 turning points 2 relative min

Slide 158 (Answer) / 276

69 How many zeros does the polynomial appear to have?

Slide 159 / 276

69 How many zeros does the polynomial appear to have?

[This object is a pull tab]

Answer

3 zeros

Slide 159 (Answer) / 276

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

Slide 160 / 276

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

[This object is a pull tab]

Answer

2 turning points 1 relative max

Slide 160 (Answer) / 276

71 How many zeros does the polynomial appear to have?

Slide 161 / 276

71 How many zeros does the polynomial appear to have?

[This object is a pull tab]

Answer

None

Slide 161 (Answer) / 276

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

Slide 162 / 276

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

[This object is a pull tab]

Answer

3 relative max 2 relative min

Slide 162 (Answer) / 276

Analyzing Graphs and Tables

• f Polynomial Functions

Return to Table of Contents

Slide 163 / 276

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.

Slide 164 / 276

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

Slide 165 / 276

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 166 / 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.

Slide 167 / 276

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.

Slide 168 / 276

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

Slide 169 / 276

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

Slide 170 / 276

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

Slide 171 / 276

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 172 / 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).

Slide 173 / 276

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.

Slide 174 / 276 Slide 175 / 276

Slide 176 / 276

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

Slide 177 / 276

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

[This object is a pull tab]

Answer

C -1 E 1

Slide 177 (Answer) / 276

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

Slide 178 / 276

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

[This object is a pull tab]

Answer

B -2 F 2

Slide 178 (Answer) / 276

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

Slide 179 / 276

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 180 / 276

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

Slide 181 / 276

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

Slide 182 / 276

Zeros and Roots of a Polynomial Function

Return to Table of Contents

Slide 183 / 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 184 / 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 185 / 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.

Slide 186 / 276

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

Slide 187 / 276

Complex Numbers Real Imaginary 3i 2 - 4i

• 0.765

2/3

• 11

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

Slide 188 / 276

Complex Numbers Real Imaginary 3i 2 - 4i

• 0.765

2/3

• 11

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

[This object is a teacher notes pull tab]

The intersection of real and imaginary should be the empty set.

Slide 188 (Answer) / 276

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)

Slide 189 / 276

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.)

Slide 190 / 276

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.

Slide 191 / 276

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

Slide 192 / 276

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

[This object is a pull tab]

Answer Zeros are -2 with multiplicity of 2 and 2 with multiplicity of 4.

Slide 192 (Answer) / 276

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.

Slide 193 / 276

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

A B

1

C

2

D

3

E

4

F

5

Slide 194 / 276

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

A B

1

C

2

D

3

E

4

F

5

[This object is a pull tab]

Answer

E 4

Slide 194 (Answer) / 276

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

Yes No

Slide 195 / 276

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

Yes No

[This object is a pull tab]

Answer

No

Slide 195 (Answer) / 276

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

A B

1

C

2

D

3

E

4

F

5

Slide 196 / 276

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

A B

1

C

2

D

3

E

4

F

5

[This object is a pull tab]

Answer C 2

Slide 196 (Answer) / 276

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

A B

1

C

2

D

3

E

4

F

5

Slide 197 / 276

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

A B

1

C

2

D

3

E

4

F

5

[This object is a pull tab]

Answer

D 3

Slide 197 (Answer) / 276

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

Yes No

Slide 198 / 276

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

Yes No

[This object is a pull tab]

Answer

Yes

Slide 198 (Answer) / 276

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

A B

1

C

2

D

3

E

4

F

5

Slide 199 / 276

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

A B

1

C

2

D

3

E

4

F

5

Answer

C

Slide 199 (Answer) / 276

89 How many imaginary zeros does this 4

th-degree

polynomial have?

A B

1

C

2

D

3

E

4

F

5

Slide 200 / 276

89 How many imaginary zeros does this 4

th-degree

polynomial have?

A B

1

C

2

D

3

E

4

F

5

[This object is a pull tab]

Answer

C 2

Slide 200 (Answer) / 276

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

A B

1

C

2

D

3

E

4

F

6

Slide 201 / 276

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

A B

1

C

2

D

3

E

4

F

6

[This object is a pull tab]

Answer F 6

Slide 201 (Answer) / 276

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

Yes No

Slide 202 / 276

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

Yes No

[This object is a pull tab]

Answer

Yes

Slide 202 (Answer) / 276

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

A B

1

C

2

D

3

E

4

F

5

Slide 203 / 276

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

A B

1

C

2

D

3

E

4

F

5

[This object is a pull tab]

Answer

C 2

Slide 203 (Answer) / 276

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: Slide 204 / 276

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

Slide 205 / 276 Slide 206 / 276

Slide 206 (Answer) / 276 Slide 207 / 276 Slide 207 (Answer) / 276

Slide 208 / 276 Slide 208 (Answer) / 276

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

• r
• r
• r
• r

Don't forget the ±!!

Slide 209 / 276

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

• r
• r
• r
• r

Don't forget the ±!!

[This object is a pull tab]

Answer This polynomial has five distinct real zeros: -6, -4, -2, 2, and 3.

• 4 and 3 each have a multiplicity of

2 (their factors are being squared) There are 2 imaginary zeros: -3i and 3i. Each with multiplicity of 1. There are 9 zeros (count -4 and 3 twice) so this is a 9th degree polynomial.

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Slide 220 (Answer) / 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.

Slide 222 / 276

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

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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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Answer

D Slide 223 (Answer) / 276

106 How many REAL zeros does the polynomial equation 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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Answer

D Slide 224 (Answer) / 276

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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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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Answer

A C E x = -2, multiplicity 1 x = 3, multiplicity 1 x = 0, multiplicity 1

Slide 225 (Answer) / 276

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

Slide 226 (Answer) / 276

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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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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Answer

A C D x = 2, multiplicity 1 x = -i, multiplicity 1 x = i, multiplicity 1

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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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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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Answer

C E F x = -2, multiplicity 1 x = , multiplicity 1 x = - , multiplicity 1

Slide 228 (Answer) / 276

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 230 / 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.

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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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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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Teacher Notes

*When you find a distinct zero, write the zero in factored form and then complete polynomial division. *This is a good time to introduce synthetic division as a means to shorten the written work.

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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 234 / 276

• 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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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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Answer

A, B

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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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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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Answer

D x = -3, multiplicity 1 B x = 1, multiplicity 2

Slide 237 (Answer) / 276

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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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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Answer

A x = -2, multiplicity 1 E x = -1, multiplicity 2

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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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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

Answer

C, E, H Slide 239 (Answer) / 276 Slide 240 / 276 Slide 240 (Answer) / 276

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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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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Answer

E, F, G, H

Slide 241 (Answer) / 276

Writing a Polynomial Function from its Given Zeros

Return to Table of Contents

Slide 242 / 276

Goals and Objectives

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

Slide 243 / 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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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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Answer

D Slide 245 (Answer) / 276

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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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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Answer

B Slide 246 (Answer) / 276

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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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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Answer

C Slide 247 (Answer) / 276

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

Slide 249 / 276

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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Answer

C Slide 249 (Answer) / 276

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

Slide 250 / 276

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

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Answer

A Slide 250 (Answer) / 276 Slide 251 / 276 Slide 251 (Answer) / 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 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

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Answer

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

y = (x + 2)

2

Slide 252 (Answer) / 276

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 253 / 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 Note: The calculator will give an estimate. Rounding may be needed.

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Answer Zeros: -2, -1.4, 1.4 Relative Max: (-1.72, 0.27) Relative Min: (0.387, -4.42)

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

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

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Answer

Zeros: -3, -1, 1, 3 Min: (-2.2, -16), (2.2, 16) Max: (0, 9)

Slide 257 (Answer) / 276

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.

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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?

Slide 259 / 276

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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Answer Equation: x(x + 1)(x + 2)(x + 3) = 175,560. Solution: x = 19, so the numbers are 19, 20, 21 and 22.

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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

Slide 260 / 276

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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Answer 1. y = x(50 - 2 x)2 2. From the table, the max appears to be about 8 (domain: 0 < x < 25). 3. Set the window at x: 0 - 25, y: 0 - 10,000. The max appears to be about 9259 at x = 8.33.

Slide 260 (Answer) / 276

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?

( (

Slide 261 / 276

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?

( (

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Answer Sample Text

Slide 261 (Answer) / 276

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

Slide 262 / 276

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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Answer x = 0, 2 and 3

Slide 262 (Answer) / 276

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?

Slide 263 / 276

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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t students discuss this question in groups or as a whole

• ss. The following conclusion should be made: To start at

ight 0 yards and end 300 yards later at height 0 yards, e multiplied x by (x - 3) (to create zeros at 0 and 3). To eate the bottom of the hill at 200 yards, she multiplied is function by (x - 2) 2 . She needed to multiply by -3 to arantee the roller coaster shape and to adjust the overall ight of the roller coaster. the function is graphed without regard to domain, the aph goes below 0, which makes no sense for a roller

• aster. Also, for x < 0, the roller coaster would be going

ckwards. e starts and stops are too abrupt for a real roller coaster. e riders would crash at the end. Real coasters flatten out the end to slow down.

Slide 263 (Answer) / 276

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 264 / 276

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?

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Answer

x-intercepts are 1, 4 and 8 At 1, 4 and 8 the function changes from pos to neg

• r neg to pos.

Slide 264 (Answer) / 276

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?

Slide 265 / 276

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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Answer

We can look at the leading coefficient and the degree to determine start and end points and the zeros in between. OR, we can evaluate the function for a point on the interval between the x- intercepts, for example, f(0) = -32, so the graph is below the x-axis for x < 1. For 1 <x < 4, f(2) = 12, so the graph is above the x-axis. For 4 < x < 8, f(5) =

• 12, so the graph is below the x-axis.

For x > 8, f(10) = 108, so the graph is above the x-axis.

Slide 265 (Answer) / 276

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

Slide 266 / 276

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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Answer

(0, 8)

Slide 266 (Answer) / 276

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

Slide 267 / 276

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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Answer

Slide 267 (Answer) / 276

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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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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Answer C

Slide 268 (Answer) / 276

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

Slide 269 / 276

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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Answer B

Slide 269 (Answer) / 276

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

Slide 270 / 276

Degree: # x-intercepts: # turning points:

Observations:

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

How many relative maxima and minima? Answer

Slide 271 / 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:

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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Answer

Something to the effect of : If y goes up as x goes up, the function is increasing. If y goes down as x goes up, the function is decreasing.

Slide 272 (Answer) / 276

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

Slide 273 / 276

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

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Answer

: increasing : decreasing

Increasing: x < -2,

• 0.2 < x <2.3,

x > 3 Decreasing:

• 2 < x < 0.2,

2.3 < x < 3

(approximately)

Slide 273 (Answer) / 276

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.

Slide 274 / 276

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.

Answer

A, C Slide 274 (Answer) / 276

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.

Slide 275 / 276

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.

Answer

A, C Slide 275 (Answer) / 276

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

Slide 276 / 276