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Algebra II

Polynomials:

Operations and Functions

www.njctl.org

2014-10-22




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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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Properties of Exponents Review

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Contents

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

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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.

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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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2 Divide:

A

B

C

D Solutions not shown

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3 Divide:

A

B

C

D Solution not shown A

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4 Simplify:

A

B

C


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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


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6 Simplify and write with positive exponents:

A

B

C

D Solution not shown A

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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


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8 Simplify the expression:

A

B

C

D

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Operations with Polynomials

ReviewReturn toTable of

Contents

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

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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

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

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Vocabulary

The degree of a polynomial is the highest exponent contained in the polynomial. When a polynomial has more than one variable, the degree is found by adding the exponents of all of the variables of each term, and finding the term with the highest value.

degreedegree=3+1+2=6

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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 order of degree from highest to lowest.

Polynomial Function

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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:

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

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

Watch your signs...forgetting to distribute the minus sign is one of 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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10 Subtract

A

B

C

D Ans

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11 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 + 1x2 +5x

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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

Multiplying Polynomials

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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?

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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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19 What is the area of the shaded region (in square units)?

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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http://www.azed.gov/azccrs/mathstandards/9-12

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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 of 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

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Contents

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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:

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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 over 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

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C

D Ans

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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-APRTrina'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.

http://illustrativemathematics.org/illustrations/594

http://illustrativemathematics.org/illustrations/594

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Binomial Theorem

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

Let's look at the final result.

Do you notice a pattern in the exponents?

The exponents of a decrease by 1, and the exponents of b increase by 1 for each term. The sum of the two exponents always equal the original power, 3.

Each term has variables and exponents where n is

the exponent from the original problem, and k is the term number beginning with k = 0:

Remember, a0 = b0 = 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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Now let's look at the coefficients.

Write only the coefficients to the right of each equation. Do you see a pattern? Can you continue the pattern to add two more rows?

We call this pattern Pascal's Triangle.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

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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

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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 of

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32 The expansion of is:

A

B

C

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33 The binomial theorem can be used to expand

True

False

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Factoring Polynomials Review

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Contents

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Factoring Polynomials ReviewThe 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

factorableE 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


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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?


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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 :


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Dividing Polynomials

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Contents

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Division of Polynomials

Here are 3 different ways to write the same quotient:

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To divide a polynomial by a monomial, write each term of the polynomial as a fraction with a common

denominator. Then simplify each fraction.

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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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Long Division of PolynomialsTo divide a polynomial by 2 or more terms, long division can be used.

Recall long division of numbers, such as 8693 ÷ 41.

or

· Multiply· Subtract· Bring down· Repeat· Write Remainder over divisor

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· Multiply· Subtract· Bring down· Repeat· Write Remainder over divisor

-2x2+-6x -10x +3 -10x -30 33

On the next several slides, we will break this down step by step....

Here is an example:

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Start by looking at the first terms. Think "what do I multiply by x to get 2x2 ?"

(Or think "2x2 / x")

Put the answer here

Continue to next slide.

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2x

Next, multiply 2x by x + 3, and put the product under the dividend. Then subtract.

- ( )


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2x

2x2 + 6x- ( )

Bring down the + 3. Repeat the whole process. This time, ask "what do I multiply by x to get -10x, or what is -10x / x?"

Put the answer here

- 10x


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2x2 + 6x- ( )- 10x + 3-10x - 30- ( )

Multiply and subtract.

33

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-10x - 30- ( )33

+ 33x + 3

2x2 + 6x- ( )- 10x + 3

Since x doesn't divide into 33, we can't divide further.

33 is the remainder, which we write as a fraction.

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Examples

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If a polynomial function f(x), of degree ≥ 1, is divided by x - a, then the remainder is equal to f(a).

The Remainder Theorem

Complete the division, then calculate f (2).

Example:

Find

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Example:

Find the quotient of

Find f (-2). What do you notice?

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Example: In this example there are "missing terms". Fill in those terms with zero coefficients before dividing. Then find f(-1).

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Example

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46 Simplify.

A

B

C

D

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47 Simplify.

A

B

C

D

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48 Divide.

A

B

C

D

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49 Divide. Students type their answers here

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50 Divide the polynomial. Students type their answers here

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51 If = ,what is f (-7)? Students type their answers here

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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 toTable of

Contents

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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.

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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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Graphs of Polynomial FunctionsFeatures:· 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 of 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

http://www.geogebratube.org/material/show/id/10583

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In analyzing graphs of functions, one feature is the "end behavior" of the graph. We want to observe and predict what happens to the graph when the value of x becomes very large or very small. This is described mathematically as "x approaches positive infinity" or "x approaches negative infinity."

End Behavior for Graphs of Functions

The notation for these expressions is

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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?

∞∞

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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

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End Behavior of a Polynomial

Lead coefficient is positive

Left End Right End

Lead coefficientis negative

Left End Right End

Polynomial of even degree

Polynomial of odd degree

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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.

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End Behavior of a PolynomialDegree: 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 or even degree, and if the lead coefficient is positive or negative.

A odd and positive

B odd and negative

C even and positive

D even and negative

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A odd and positive

B odd 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 = 7x0)

Examples:

Odd function Even function Neither

f(x)=3x5 - 4x3 + 2x h(x)=6x4 - 2x2 + 3 g(x)= 3x2 + 4x - 4

y = 5x y = x2 y = 6x - 2

g(x)=7x7 + 2x3 f(x)=3x10 -7x2 r(x)= 3x5 +4x3 -2

Odd and Even Functions

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58 Which of the following are odd functions? (Select all that apply.)

A

B

C

D

E

F

G

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59 Which of the following are even functions? (Select all that apply.)

A

B

C

D

E

F

G

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An odd function has rotational symmetry about the origin, (0,0). If you spin about the origin 180°, the graph will be the same. (Think of a pin-wheel.)

Definition of an Odd Function

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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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A Odd Degree

B Odd Function

C Even Degree

D Even Function



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

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

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Analyzing Graphs and Tables of Polynomial Functions

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x y-3 58

-2 19

-1 0

0 -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 0

0 -5

1 -2

2 3

3 4

4 -5

How many zeros does this function appear to have?

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x y-3 58

-2 19

-1 0

0 -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?

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Intermediate Value TheoremGiven 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 0

0 -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

0 3

1 0

2 -2

3 4

4 -5

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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 0

E 1

F 2

G 3

H 4

x y-3 -12

-2 -4

-1 1

0 3

1 0

2 -2

3 4

4 -5

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

x y-3 2

-2 0

-1 5

0 2

1 -3

2 4

3 4

4 -5

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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 0

E 1

F 2

G 3

H 4

x y-3 2

-2 0

-1 5

0 2

1 -3

2 4

3 4

4 -5

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Relative Maxima and Relative MinimaThere 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 maxima and minima from a table?

x f(x)-3 5

-2 1

-1 -1

0 -4

1 -5

2 -2

3 2

4 0

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 as x goes from 3 to 4, f(x)is decreasing.

The relative maxima and minima occur 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 or falls.

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When y switches from increasing to decreasing there is a maximum. What is the approximate value of x for the relative max?

x y-3 5

-2 1

-1 -1

0 -4

1 -5

2 -2

3 2

4 0

Relative Max:

click to reveal

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When y switches from decreasing to increasing there is a minimum. What is the approximate value of x for the relative min?

x y-3 5

-2 1

-1 -1

0 -4

1 -5

2 -2

3 2

4 0

Relative Min:

click to reveal

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

A -3

B -2

C -1

D 0

E 1

F 2

G 3

H 4

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78 At about what x-values does a relative maximum occur?

A -3

B -2

C -1

D 0

E 1

F 2

G 3

H 4

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79 At about what x-values does a relative minimum occur?

A -3

B -2

C -1

D 0

E 1

F 2

G 3

H 4

x y-3 5

-2 1

-1 -1

0 -4

1 -5

2 -2

3 2

4 0

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H 4

x y-3 5

-2 1

-1 -1

0 -4

1 -5

2 -2

3 2

4 0

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81 At about what x-values does a relative minimum occur?

A -3

B -2

C -1

D 0

E 1

F 2

G 3

H 4

x y-3 2

-2 0

-1 5

0 2

1 -3

2 4

3 4

4 -5

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H 4

x y-3 2

-2 0

-1 5

0 2

1 -3

2 4

3 5

4 -5

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Zeros and Roots of a Polynomial Function

Return toTable of

Contents

page1svg

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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).

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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.

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Complex NumbersComplex 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 -119+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 on 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 occurs 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 0

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 0

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 0

B 1

C 2

D 3

E 4

F 5

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Yes

No

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

A 0

B 1

C 2

D 3

E 4

F 5

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89 How many imaginary zeros does this 4th-degree polynomial have?

A 0

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 0

B 1

C 2

D 3

E 4

F 6

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Yes

No

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

A 0

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.

What is the degree of the polynomial?

or or or

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93 How many distinct real zeros does this polynomial have?

A 0

B 1

C 2

D 3

E 4

F 5

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94 What are the real zeros of

A -4

B -3

C -2

D -1E 0

F 1G 2

H 3

I 4

?

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or or or or

Don't forget the ±!!

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Find the zeros, both real and imaginary, showing the multiplicities, of the following polynomial:

This polynomial has1 real root: 2and 2 imaginary roots:-1i and 1i. They are simple roots with multiplicities of 1.

click to reveal

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95 What is the multiplicity of the root x = 1?

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96 How many distinct real zeros does the polynomial have?

A 0

B 1

C 2

D 3

E 4

F 5

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97 How many distinct imaginary zeros does the polynomial have?

A 0

B 1

C 2

D 3

E 4

F 5

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98 What is the multiplicity of x = 1?

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99 What are the distinct real zeros of the polynomial?

A -4

B 4

C 1

D

E -7

F 7

G 0

H 2

I -2-1

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100 What are the imaginary roots of the polynomial?

A i

B -i

C 2i

D -2i

E 7i

F -7i

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102 How many distinct real zeros does the polynomial have?

A 0

B 5

C 6

D 7

E 8

F 9

Ans

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104 How many distinct imaginary zeros does the polynomial have?

A 0

B 1

C 2

D 3

E 4

F 5

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

or or

or or

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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or

or

or

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?

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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 THEOREMMake 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:

or or

or or

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:

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or or

or or

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


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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A x = -2, multiplicity 1

B x = -2, multiplicity 2

C x = -2, multiplicity 3


E x = -1, multiplicity 2

F x = -1, multiplicity 3

Pul

l fo

r A

ns wer

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B x = -1, multiplicity 1



E x = , multiplicity 1

F x = , multiplicity 1

G x = , multiplicity 1

H x = , multiplicity 1

Pul

l for

A

nsw

er

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115 Use the Rational Zeros Theorem to find the zeros of the polynomial equation, including multiplicities.

A x = 3, mulitplicity 1

B x = 2, mulitplicity 2



E x = , multiplicity 1

F x = , multiplicity 2

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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

Return toTable of

Contents

page1svg

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Goals and Objectives· Students will be able to write a polynomial

from its given zeros.

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Write (in factored form) the polynomial function of lowest degree using the given zeros, including any multiplicities.

x = -1, multiplicity of 1x = -2, multiplicity of 2x = 4, multiplicity of 1

or or or

or or or

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 1x = 3, multiplicity of 1x = 2.5, multiplicity of 1

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A

B

C

D

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

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A

B

C

D

E

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

or or or

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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

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122 Which equation could be the equation of the graph below?

A

B

C

D

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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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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

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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.

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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?

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Derived from( (

https://www.engageny.org/resource/high-school-algebra-ii




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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?

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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?


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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.

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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


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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


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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)

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Degree:

# x-intercepts:

# turning points:

Observations:

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

How many relative maxima and minima?

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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

on 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)=-x3 + 5

Even Odd Neither

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

k(x)

k(x) Ans

wer



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