Unit 1- Polynomial Functions

Common Core Algebra II

Unit 1- Polynomial Functions

Lesson 1- The Multiplication of Polynomials

1

Lesson 2- The Division of Polynomials

5

Lesson 3- Long Division, Again?

9

Lesson 4- Operations with Polynomials

14

Lesson 5- Polynomial Identities 18

Lesson 6- GCF and The Difference of Squares

22

Lesson 7- Factoring Perfect Cubes 26

Lesson 8- Factoring Trinomial Review

30

Lesson 9- Seeing Structure: Grouping

33

Lesson 10- Seeing Structure: Advance Factoring

38

Lesson 11- The Special Role of Zero in Factoring

41

Lesson 12- Graphing Factored Polynomials

46

Lesson 13- End Behavior of Polynomials

51

Lesson 14- Even and Odd Functions

57

Lesson 15- Modeling with Polynomial Functions 61

Lesson 16- What If There is a Remainder?

64

Lesson 17- The Remainder Theorem

67

Lesson 18- Putting It All Together 71

Common Core Algebra II

Unit 1 Common Core State Standards • A.SSE.A.2- Use the structure of an expression to identify ways to rewrite it. For

example, factor expressions involving GCF, difference of squares, perfect cubes, trinomials, and grouping.

• A.APR.B.2- Know and apply the Remainder Theorem: For a polynomial 𝑝𝑝(𝑥𝑥) and a number 𝑎𝑎, the remainder on division by 𝑥𝑥 − 𝑎𝑎 is 𝑝𝑝(𝑎𝑎), so 𝑝𝑝(𝑎𝑎) = 0 if and only if 𝑥𝑥 − 𝑎𝑎 is a factor of 𝑝𝑝(𝑥𝑥).

• A.APR.B.3- Identify zeros of polynomials when suitable factorizations are

available, and use the zeros to construct a rough graph of the function defined by the polynomial.

• A.APR.C.4- Prove polynomial identities and use them to describe numerical

relationships.

• A.APR.D.6- Rewrite simple rational expressions in different forms; write 𝑎𝑎(𝑥𝑥)/𝑏𝑏(𝑥𝑥) in the form 𝑞𝑞(𝑥𝑥) + 𝑟𝑟(𝑥𝑥)/𝑏𝑏(𝑥𝑥), where 𝑎𝑎(𝑥𝑥), 𝑏𝑏(𝑥𝑥), 𝑞𝑞(𝑥𝑥), and 𝑟𝑟(𝑥𝑥) are polynomials with the degree of 𝑟𝑟(𝑥𝑥) less than the degree of 𝑏𝑏(𝑥𝑥), using inspection or long division.

• F.IF.C.7c- Graph polynomial functions, identifying zeros when suitable

factorizations are available, and showing end behavior.

• F.BF.B.3- Recognizing even and odd functions from their graphs and algebraic expressions for them.

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 1: The Multiplication of Polynomials

1

Opening Exercise

Show that 28 × 27 = (20 + 8)(20 + 7) using an area model.

1. Use the tabular method to multiply (𝑥𝑥 + 8)(𝑥𝑥 + 7) and combine like terms. Explain how the result is related to 756 from the Opening Exercise.

How can we multiply these binomials without using a table?

Common Core Algebra II Lesson 1: The Multiplication of Polynomials

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2. Use the tabular method to multiply (𝑥𝑥2 + 3𝑥𝑥 + 1)(𝑥𝑥2 − 5𝑥𝑥 + 2) and combine like terms.

3. Use the tabular method to multiply (𝑥𝑥2 + 3𝑥𝑥 + 1)(𝑥𝑥2 − 2) and combine like terms.

4. Using the distributive property, express the product (𝑥𝑥 − 1)(𝑥𝑥2 + 𝑥𝑥 + 1) in standard form.

Common Core Algebra II Lesson 1: The Multiplication of Polynomials

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5. Find the product (𝑥𝑥 − 1)(𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1) using table.

6. Using exercises 3 & 4, generalize the pattern that emerges by writing an identity for (𝑥𝑥 − 1)(𝑥𝑥𝑛𝑛 + 𝑥𝑥𝑛𝑛−1 + ⋯+ 𝑥𝑥2 + 𝑥𝑥 + 1) for positive integer 𝑛𝑛.

7. Create an equivalent expression to (𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐)2.

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 1: The Multiplication of Polynomials

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Homework Multiply the following polynomials and express your answer in standard form.

1. (𝑥𝑥2 − 4𝑥𝑥 + 4)(𝑥𝑥 + 3) 2. (2𝑥𝑥 − 3)(𝑥𝑥3 + 𝑥𝑥2 + 𝑥𝑥 + 1)

3. (𝑥𝑥2 − 3𝑥𝑥 + 9)(𝑥𝑥2 + 3𝑥𝑥 + 9) 4. (𝑡𝑡 + 1)(𝑡𝑡 − 1)(𝑡𝑡2 + 1)

5. If 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 + 1 and 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥2 + 1, express 𝑓𝑓(𝑥𝑥) ⋅ 𝑔𝑔(𝑥𝑥) in standard form.

Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 2: The Division of Polynomials

5

Opening Exercise

Multiply these polynomials using the tabular method.

(2𝑥𝑥 + 5)(𝑥𝑥2 + 5𝑥𝑥 + 1)

How can you use your answer from above to quickly multiply 25 ∙ 151?

1. Show that 2𝑥𝑥3+15𝑥𝑥2+27𝑥𝑥+5

2𝑥𝑥+5 = 𝑥𝑥2 + 5𝑥𝑥 + 1

Common Core Algebra II Lesson 2: The Division of Polynomials

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2. Reverse the tabular of multiplication to find the quotient 2𝑥𝑥2+𝑥𝑥−10

𝑥𝑥−2.

3. Create your own table and use the 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑟𝑟 𝑚𝑚𝑟𝑟𝑡𝑡ℎ𝑜𝑜𝑜𝑜 to find the quotient.

𝑥𝑥4 + 4𝑥𝑥3 + 3𝑥𝑥2 + 4𝑥𝑥 + 2𝑥𝑥2 + 1

Explain how we can use the previous quotient to factor 𝑥𝑥4 + 4𝑥𝑥3 + 3𝑥𝑥2 + 4𝑥𝑥 + 2.

Common Core Algebra II Lesson 2: The Division of Polynomials

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4. Use the 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑟𝑟 𝑚𝑚𝑟𝑟𝑡𝑡ℎ𝑜𝑜𝑜𝑜 to find the quotient.

3𝑥𝑥5 − 2𝑥𝑥4 + 6𝑥𝑥3 − 4𝑥𝑥2 − 24𝑥𝑥 + 16𝑥𝑥2 + 4

OYO!

5. Find the following quotients:

4𝑥𝑥3−10𝑥𝑥2−22𝑥𝑥−82𝑥𝑥+1

𝑥𝑥4+3𝑥𝑥3−6𝑥𝑥2−6𝑥𝑥+8𝑥𝑥+4

Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 2: The Division of Polynomials

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Homework 1. Use the reverse tabular method to find the following quotients.

2𝑥𝑥3+𝑥𝑥2−16𝑥𝑥+152𝑥𝑥−3

𝑥𝑥3−8𝑥𝑥−2

3𝑥𝑥5+12𝑥𝑥4+11𝑥𝑥3+2𝑥𝑥2−4𝑥𝑥−23𝑥𝑥2−1

4𝑥𝑥2+8𝑥𝑥+32𝑥𝑥+1

2. First compute 3𝑥𝑥3+10𝑥𝑥2−14𝑥𝑥+4

3𝑥𝑥−2. Then express 3𝑥𝑥3 + 10𝑥𝑥2 − 14𝑥𝑥 + 4 as the product of

two polynomials. Explain your reasoning.

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 3: Long Division, Again?

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Opening Exercise

Use the reverse tabular method to determine the quotient 2𝑥𝑥3+11𝑥𝑥2+7𝑥𝑥+10

𝑥𝑥+5.

Write the polynomial 2𝑥𝑥3 + 11𝑥𝑥2 + 7𝑥𝑥 + 10 in factored form.

1. Take a trip back to elementary school and use long division to evaluate 1573 ÷ 13.

Common Core Algebra II Lesson 3: Long Division, Again?

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2. Let’s return to back to Algebra 2. If we let 𝑥𝑥 = 10, then we can represent the previous problem as

3753 23 ++++ xxxx

3. Use the long division algorithm for polynomials to evaluate 2𝑥𝑥3 − 4𝑥𝑥2 + 2

2𝑥𝑥 − 2.


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4. Use the long division algorithm to determine the quotient. For each problem, check your work by using the reverse tabular method or using multiplication.

a. 7𝑥𝑥3−8𝑥𝑥2−13𝑥𝑥+2

7𝑥𝑥−1 b. 𝑥𝑥

2+6𝑥𝑥+9𝑥𝑥+3

c. 𝑥𝑥3−27𝑥𝑥−3

d. 2𝑥𝑥4+14𝑥𝑥3+𝑥𝑥2−21𝑥𝑥−6

2𝑥𝑥2−3


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e. 5𝑥𝑥4−6𝑥𝑥2+1

𝑥𝑥2−1 f.

𝑥𝑥3+2𝑥𝑥2+4𝑥𝑥+8𝑥𝑥+2

g. 2𝑥𝑥7+𝑥𝑥5−4𝑥𝑥3+14𝑥𝑥2−2𝑥𝑥+7

2𝑥𝑥2+1 h.

𝑥𝑥6−64𝑥𝑥+2

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 3: Long Division, Again?

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Homework 1. Use the long division algorithm to determine the quotients.

2𝑥𝑥3−13𝑥𝑥2−𝑥𝑥+32𝑥𝑥+1

3𝑥𝑥3+4𝑥𝑥2+7𝑥𝑥+22

𝑥𝑥+2

2. Given 𝑓𝑓(𝑥𝑥) = 4𝑥𝑥3 + 5𝑥𝑥 + 21 and ℎ(𝑥𝑥) = 2𝑥𝑥 + 3, express 𝑓𝑓(𝑥𝑥)ℎ(𝑥𝑥)

in standard form by using

long division.

3. Given 𝑞𝑞(𝑥𝑥) = 3𝑥𝑥3 − 4𝑥𝑥2 + 5𝑥𝑥 + 𝑘𝑘, determine the value of 𝑘𝑘 so that 3𝑥𝑥 − 7 is a factor of the polynomial 𝑞𝑞. Explain your reasoning.

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 4: Operations with Polynomials

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Opening Exercise

Find the sum of 3𝑥𝑥2 − 7𝑥𝑥 + 11 and 5𝑥𝑥3 − 22𝑥𝑥2 + 9𝑥𝑥 − 5.

Find the difference 3𝑥𝑥2 − 7𝑥𝑥 + 11 − (5𝑥𝑥3 − 22𝑥𝑥2 + 9𝑥𝑥 − 5).

1. We can combine operations together. Rewrite each polynomial in standard form by applying the operations in the appropriate order.

a. (𝑥𝑥2+5𝑥𝑥+20)+(𝑥𝑥2+6𝑥𝑥−6)

𝑥𝑥+2 b. (𝑥𝑥2 − 4)(𝑥𝑥 + 2) − 3(𝑥𝑥2 + 2𝑥𝑥 − 5)

2. A manufacture has developed a cost model, 𝐶𝐶(𝑥𝑥) = 0.15𝑥𝑥3 + 0.01𝑥𝑥2 + 2𝑥𝑥 + 120, where 𝑥𝑥 is the number of items sold, in thousands. The sales price can be modeled by 𝑆𝑆(𝑥𝑥) = 30 − 0.01𝑥𝑥. Therefore, revenue is modeled by 𝑅𝑅(𝑥𝑥) = 𝑥𝑥 ⋅ 𝑆𝑆(𝑥𝑥).

Write a polynomial in standard form that can be used to model the company’s profits 𝑃𝑃(𝑥𝑥) = 𝑅𝑅(𝑥𝑥) − 𝐶𝐶(𝑥𝑥).

Common Core Algebra II Lesson 4: Operations with Polynomials

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Polynomial Pass

Use the next two pages to complete the exercise on the index cards. You will then pass your index card after two minutes and receive a new problem. The answer to the problem you just completed will be on the back of your new card. Make sure you pass the cards in order!

Common Core Algebra II Lesson 4: Operations with Polynomials

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More space to work!

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 4: Operations with Polynomials

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Homework 1. Perform the indicated operations to write each polynomial in standard form.

(2𝑥𝑥2 − 𝑥𝑥3 − 9𝑥𝑥 + 1) − 4(𝑥𝑥3 + 7𝑥𝑥 − 3𝑥𝑥2 + 1) (𝑥𝑥 + 3)2 − (𝑥𝑥 + 4)2

𝑥𝑥2−5𝑥𝑥+6𝑥𝑥−3

+𝑥𝑥2 + 𝑥𝑥 + 1 (𝑥𝑥 + 3)(𝑥𝑥 − 3) − (𝑥𝑥 + 4)(𝑥𝑥 − 4)

2. What is the area of the figure below? Assume there is a right angle at each vertex.

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 5: Polynomial Identities

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Opening Exercise

Show that the sum of three consecutive integers is the three times the middle integer.

1. Prove that if 𝑥𝑥 > 1, then a triangle with side lengths 𝑥𝑥2 − 1, 2𝑥𝑥, and 𝑥𝑥2 + 1 is a right triangle.

Pick a value of 𝑥𝑥 to create a Pythagorean triple.

Explain why every Pythagorean triple must contain an even integer.

Common Core Algebra II Lesson 5: Polynomial Identities

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2. Prove that (𝑥𝑥 + 𝑎𝑎)2 = 𝑥𝑥2 + 2𝑎𝑎𝑥𝑥 + 𝑎𝑎2 is an identity.

3. Use this identity to quickly compute the following expressions.

(𝑥𝑥 + 5)2 (3𝑦𝑦 − 4)2 (5𝑥𝑥 + 2𝑦𝑦)2

4. Prove that the difference of the squares of any two consecutive integers is always odd.

Common Core Algebra II Lesson 5: Polynomial Identities

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5. Algebraically determine the values of 𝑎𝑎 and 𝑏𝑏 to correctly complete the identity stated below.

4𝑥𝑥3 + 𝑎𝑎𝑥𝑥2 + 23𝑥𝑥 + 20 = (2𝑥𝑥 + 5)(2𝑥𝑥2 + 𝑏𝑏𝑥𝑥 + 4)

6. Prove the identity (𝑎𝑎2 + 𝑏𝑏2)(𝑥𝑥2 + 𝑦𝑦2) = (𝑎𝑎𝑥𝑥 − 𝑏𝑏𝑦𝑦)2 + (𝑏𝑏𝑥𝑥 + 𝑎𝑎𝑦𝑦)2.

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 5: Polynomial Identities

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Homework 1. Prove that (𝑥𝑥 + 𝑦𝑦)2 − (𝑥𝑥 − 𝑦𝑦)2 = 4𝑥𝑥𝑦𝑦 for all real numbers 𝑥𝑥 and 𝑦𝑦.

2. Prove that (𝑚𝑚 + 𝑛𝑛)3 = 𝑚𝑚3 + 3𝑚𝑚2𝑛𝑛 + 3𝑚𝑚𝑛𝑛2 + 𝑛𝑛3 is an identity.

3. The identity (𝑥𝑥2 + 𝑦𝑦2)2 = (𝑥𝑥2 − 𝑦𝑦2)2 + (2𝑥𝑥𝑦𝑦)2 can be used to generate Pythagorean triples. Show that this statement is an identity.

Pick values for 𝑥𝑥 and 𝑦𝑦, where 𝑥𝑥 > 𝑦𝑦, to generate a Pythagorean triple.

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 6: GCF and The Difference of Squares

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Opening Exercise

Prove the polynomial identity 𝑎𝑎2 − 𝑏𝑏2 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑏𝑏).

Back in Algebra I, we factored expression by undoing the distributive property. This process is known as factoring out the greatest common factor (GCF).

1. Factor the following expression by factoring out the GCF.

6𝑥𝑥2 + 18𝑥𝑥 + 10 3𝑥𝑥3 + 18𝑥𝑥

2𝑑𝑑4 + 6𝑑𝑑3 − 18𝑑𝑑2 − 54𝑑𝑑 5𝑥𝑥3𝑦𝑦 − 30𝑥𝑥2𝑦𝑦2 − 3𝑥𝑥𝑦𝑦3

Sometimes the GCF can be more than just monomial!

2. Factor out the GCF from each of the following expression.

2𝑥𝑥(𝑥𝑥 + 5) − 3(𝑥𝑥 + 5) 𝑥𝑥2(3𝑥𝑥 + 5) + 16(3𝑥𝑥 + 5)

𝑡𝑡(𝑡𝑡2 + 5𝑡𝑡 + 6) − 2(𝑡𝑡2 + 5𝑡𝑡 + 6) 𝑘𝑘2(𝑘𝑘 + 4) + 8𝑘𝑘(𝑘𝑘 + 4) + 12(𝑘𝑘 + 4)

Common Core Algebra II Lesson 6: GCF and The Difference of Squares

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We also factored the difference of squares in Algebra I.

3. Factor the following expression completely.

𝑥𝑥2 − 25 1 − 9𝑦𝑦2 81𝑦𝑦4 − 16𝑥𝑥4

What about the sum of perfect squares? It’s reasonable to think that 𝑥𝑥 + 𝑎𝑎 or 𝑥𝑥 − 𝑎𝑎 could be a factor of 𝑥𝑥2 + 𝑎𝑎2.

4. Compute the quotients below if possible.

𝑥𝑥2 + 4𝑥𝑥 + 2

𝑥𝑥2 + 4𝑥𝑥 − 2

5. Factor the expressions completely.

𝑦𝑦 − 𝑦𝑦5 2𝑥𝑥3 − 18𝑥𝑥 𝑥𝑥8 − 1

Common Core Algebra II Lesson 6: GCF and The Difference of Squares

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6. The expression 𝑥𝑥2(𝑥𝑥 − 𝑦𝑦)3 − 𝑦𝑦2(𝑥𝑥 − 𝑦𝑦)3 can be rewritten as (𝑥𝑥 + 𝑦𝑦)(𝑥𝑥 − 𝑦𝑦)𝑎𝑎. Determine and state the value of 𝑎𝑎.

7. Factor the expression (𝑥𝑥 − 1)2 − 4 as the difference of perfect squares.

8. Factor the expression 𝑎𝑎2(𝑥𝑥4 − 𝑦𝑦4) − 4(𝑥𝑥4 − 𝑦𝑦4) completely.

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 6: GCF and The Difference of Squares

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Homework 1. Factor the expressions completely.

𝑥𝑥4 − 81 9𝑘𝑘2 − 49

2. Factor, over the integers, the expression 12𝑡𝑡8 − 75𝑡𝑡4 completely.


4𝑥𝑥2(𝑥𝑥 + 5) − 9(𝑥𝑥 + 5) 𝑥𝑥2(6𝑥𝑥 − 5𝑦𝑦) − 4𝑦𝑦2(6𝑥𝑥 − 5𝑦𝑦)

4. Explain why the expression 𝑥𝑥2 + 1 cannot be factored over the real numbers as (𝑥𝑥 + 1)(𝑥𝑥 − 1).

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 7: Perfect Cubes

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Opening Exercise

Write out the list of perfect cubes from 1 to 6.

Explain why we call 𝑥𝑥3 − 𝑎𝑎3 the difference of perfect cubes.

1. Find the quotient of 𝑥𝑥3−𝑎𝑎3

𝑥𝑥−𝑎𝑎. Explain how you can use this to factor 𝑥𝑥3 − 𝑎𝑎3.

2. Factor the expression completely.

𝑥𝑥3 − 27 125𝑦𝑦3 − 1 2𝑧𝑧4 − 16𝑧𝑧

Common Core Algebra II Lesson 7: Perfect Cubes

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While we could not factor the sum of perfect squares (currently), maybe we will be more successful factoring the sum of perfect cubes.

3. Find the quotient of 𝑥𝑥3+𝑎𝑎3

𝑥𝑥+𝑎𝑎. Explain how you can use this to factor 𝑥𝑥3 + 𝑎𝑎3.

4. Factor the following expressions completely.

𝑛𝑛3 + 216 2𝑥𝑥5 + 128𝑥𝑥2 27𝑥𝑥3 + 8𝑧𝑧3

5. Factor the expression 𝑥𝑥6 − 1 completely.

Common Core Algebra II Lesson 7: Perfect Cubes

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OYO

6. Factor each of the following expressions completely.

𝑥𝑥3 − 125 𝑥𝑥3 − 216𝑦𝑦3

8𝑥𝑥4 + 𝑥𝑥 𝑥𝑥3(𝑥𝑥 + 4) + 64(𝑥𝑥 + 4)

𝑎𝑎3 − 8𝑏𝑏3 128𝑥𝑥4 + 54𝑥𝑥𝑦𝑦3

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 7: Perfect Cubes

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Homework Factor the expressions completely.

𝑥𝑥3 + 8 1 − 𝑥𝑥3

792𝑥𝑥6 + 64𝑦𝑦6 𝑥𝑥3 − 𝑥𝑥

125𝑧𝑧3 + 1 𝑎𝑎4 − 𝑏𝑏4

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 8: Factoring Trinomial Review

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Opening Exercise

Find the product of (𝑥𝑥 + 6)(𝑥𝑥 + 3).

Factor the following polynomials completely.

𝑥𝑥2 + 8𝑥𝑥 + 15 3𝑥𝑥2 + 12𝑥𝑥 − 15 −𝑥𝑥3 + 12𝑥𝑥2 − 20𝑥𝑥

The 𝑎𝑎 ∙ 𝑐𝑐 Method

Factor the trinomial 2𝑥𝑥2 + 11𝑥𝑥 + 12.

Common Core Algebra II Lesson 7: Factoring Trinomial Review

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Use the 𝑎𝑎 ∙ 𝑐𝑐 method to factor the trinomials below.

a. 3𝑥𝑥2 + 17𝑥𝑥 − 6 b. 4𝑥𝑥2 + 4𝑥𝑥 − 15

c. 6𝑥𝑥2 + 7𝑥𝑥 − 20 d. −3𝑥𝑥2 − 5𝑥𝑥 + 2

e. 6𝑥𝑥2 − 11𝑥𝑥 + 3 f. 16𝑥𝑥2 − 8𝑥𝑥 − 3

Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 8: Factoring Trinomial Review

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Homework Factoring the following polynomials completely.

6𝑥𝑥2 + 48𝑥𝑥 + 90 3𝑥𝑥2 + 4𝑥𝑥 − 20

4𝑡𝑡2 + 25𝑡𝑡 + 25 5𝑥𝑥3 − 41𝑥𝑥2 + 8𝑥𝑥

8𝑚𝑚2 + 20𝑚𝑚 − 12 9𝑥𝑥4 + 35𝑥𝑥2 − 4

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 9: Seeing Structure: Grouping

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Opening Exercise

Factor the trinomial 8𝑥𝑥2 − 10𝑥𝑥 + 3.

1. Factor the expression below completely by grouping the terms.

𝑥𝑥3 − 5𝑥𝑥2 − 4𝑥𝑥 + 20

2. Use grouping to factor the cubic polynomials below completely if it can be factored.

𝑥𝑥3 − 8𝑥𝑥 + 2𝑥𝑥 − 16 𝑥𝑥3 + 2𝑥𝑥2 − 𝑥𝑥 + 2 4𝑥𝑥3 + 2𝑥𝑥2 − 36𝑥𝑥 − 18

Common Core Algebra II Lesson 9: Seeing Structure: Grouping

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3. This polynomial isn’t cubic, but maybe we can still use grouping to factor it completely.

𝑥𝑥4 + 5𝑥𝑥3 + 8𝑥𝑥 + 40

4. The concept of “grouping” can be used to factor some interesting expressions.

𝑝𝑝4 − 4𝑝𝑝2 + 5𝑝𝑝3 − 20𝑝𝑝 + 6𝑝𝑝2 − 24

𝑥𝑥5 + 5𝑥𝑥4 + 4𝑥𝑥3 + 𝑥𝑥2 + 5𝑥𝑥 + 4


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5. Express 6𝑚𝑚3−30𝑚𝑚2+4𝑚𝑚2−20𝑚𝑚−2𝑚𝑚+10

𝑚𝑚+1 as the product of linear factors.

We have learned a variety of factoring strategies this in this unit. Let’s organize our techniques of factoring polynomials.

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 9: Seeing Structure: Grouping

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Homework 1. Factor the following polynomials completely.

𝑥𝑥3 − 6𝑥𝑥2 − 25𝑥𝑥 + 150 3𝑥𝑥3 − 5𝑥𝑥2 − 48𝑥𝑥 + 80

𝑥𝑥4 − 4𝑥𝑥3 + 4𝑥𝑥2 − 16𝑥𝑥 4𝑥𝑥3 + 2𝑥𝑥2 − 36𝑥𝑥 − 18

2. Explain why 𝑥𝑥3 + 3𝑥𝑥2 − 2𝑥𝑥 + 6 cannot be factored using grouping.


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6𝑥𝑥3 − 5𝑥𝑥2𝑦𝑦 − 24𝑥𝑥𝑦𝑦2 + 20𝑦𝑦3

𝑥𝑥3 − 3𝑥𝑥2 − 6𝑥𝑥2 + 18𝑥𝑥 + 8𝑥𝑥 − 24

2𝑥𝑥4 − 7𝑥𝑥3 − 15𝑥𝑥2 − 8𝑥𝑥2 + 28𝑥𝑥 + 60

Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 10: Seeing Structure: Advance Factoring

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Opening Exercise

Factor the polynomial 4𝑥𝑥5 − 𝑥𝑥4 − 4𝑥𝑥3 + 𝑥𝑥2 completely.

Today’s main goal is to “Look for and make use of structure” in expressions. We will be factoring polynomials that may look frightening, but if we take a step back we will see the expression looks familiar.


a. 𝑥𝑥2 + 4𝑥𝑥 + 3 b. (2𝑥𝑥 + 1)2 + 4(2𝑥𝑥 + 1) + 3

c. 𝑥𝑥2 − 13𝑥𝑥 + 36 d. 𝑥𝑥4 − 13𝑥𝑥2 + 36

Common Core Algebra II Lesson 10: Seeing Structure: Advance Factoring

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e. 𝑥𝑥2 + 5𝑥𝑥 − 6 f. (𝑥𝑥3 + 2)2 + 5(𝑥𝑥3 + 2) − 6

OYO!

As usual, factor completely.

(𝑥𝑥2 + 4𝑥𝑥)2 − 16 𝑦𝑦6 − 7𝑦𝑦3 − 8

(3𝑥𝑥2 − 𝑥𝑥)2 − 32(3𝑥𝑥2 − 𝑥𝑥) + 60

Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 10: Seeing Structure: Advance Factoring

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Homework Factor the following polynomials completely.

1. 𝑦𝑦4 − 21𝑦𝑦2 − 100 2. 𝑛𝑛6 − 𝑛𝑛4 − 16𝑛𝑛2 + 16

3. (2𝑥𝑥2 − 7𝑥𝑥)2 − (2𝑥𝑥2 − 7𝑥𝑥) − 12 4. 8𝑥𝑥6 + 7𝑥𝑥3 − 1

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 11: The Special Role of Zero in Factoring

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Opening Exercise

For each equation, list some possible values of 𝑥𝑥 and 𝑦𝑦.

𝑥𝑥𝑦𝑦 = 10 𝑥𝑥𝑦𝑦 = 1 𝑥𝑥𝑦𝑦 = −1 𝑥𝑥𝑦𝑦 = 0

Does one equation tell you more information than the others?

The Zero Product Property:

1. Find all solutions to the equation (𝑥𝑥2 + 5𝑥𝑥 + 6)(𝑥𝑥2 − 3𝑥𝑥 − 4) = 0.

2. Find all solutions to the equation (𝑥𝑥3 − 9𝑥𝑥)(𝑥𝑥3 + 𝑥𝑥2 − 𝑥𝑥 − 1) = 0

Common Core Algebra II Lesson 11: The Special Role of Zero in Factoring

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3. Suppose we know that the polynomial equation 2𝑥𝑥3 + 9𝑥𝑥2 + 𝑥𝑥 − 12 = 0 has three real solutions and that one of the factors of 2𝑥𝑥3 + 9𝑥𝑥2 + 𝑥𝑥 − 12 is 𝑥𝑥 − 1. How can we find all three solutions to the given equation?

Let’s look at this the polynomial 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥3 + 9𝑥𝑥2 + 𝑥𝑥 − 12 graphically.

Factor-Zero Theorem:


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4. Consider the polynomial functions 𝑝𝑝(𝑥𝑥) = (𝑥𝑥 − 2)(𝑥𝑥 + 3)2, 𝑞𝑞(𝑥𝑥) = (𝑥𝑥 − 2)2(𝑥𝑥 + 3)4, and 𝑟𝑟(𝑥𝑥) = (𝑥𝑥 − 2)4(𝑥𝑥 − 3)5.

Quickly, find the zeros of all three functions.

5. Find the zeros of the following polynomial functions, with their multiplicities.

a. 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 + 1)(𝑥𝑥 − 1)(𝑥𝑥2 + 1) b. 𝑔𝑔(𝑥𝑥) = (𝑥𝑥 − 4)3(𝑥𝑥 − 2)8

c. ℎ(𝑥𝑥) = (2𝑥𝑥 − 3)5 d. 𝑘𝑘(𝑥𝑥) = (3𝑥𝑥 + 4)100(𝑥𝑥 − 17)4

6. Find a polynomial function that has the following zeros and multiplicities. What is the degree of your polynomial? Is it the only one?

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 11: The Special Role of Zero in Factoring

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Homework

1. Find all real solutions to the given equations.

(𝑥𝑥 − 5)(3𝑥𝑥 + 2)(𝑥𝑥 + 3) = 0 (4𝑥𝑥2 − 9)(𝑥𝑥2 − 16) = 0

6𝑥𝑥3 − 27𝑥𝑥2 − 15𝑥𝑥 = 0 𝑥𝑥3 + 3𝑥𝑥2 − 4𝑥𝑥 = 12


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2. Determine all real zeros and state their multiplicity for the function below.

𝑝𝑝(𝑥𝑥) = (𝑥𝑥2 − 16)(𝑥𝑥3 + 4𝑥𝑥2 − 16𝑥𝑥 − 64)

3. Suppose we know that 𝑥𝑥 + 2 is a factor of 𝑝𝑝(𝑥𝑥) = 4𝑥𝑥3 + 12𝑥𝑥2 + 5𝑥𝑥 − 6. Find the other factors of 𝑝𝑝 and use them to determine when 𝑝𝑝(𝑥𝑥) = 0.

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 12: Graphing Factored Polynomials

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Opening Exercise

Find algebraically the zeros of 𝑝𝑝(𝑥𝑥) = 𝑥𝑥3 − 𝑥𝑥2 − 4𝑥𝑥 + 4.

On the set of axes below, graph 𝑦𝑦 = 𝑝𝑝(𝑥𝑥).

Explain what the zeros represent on the graph of 𝑦𝑦 = 𝑝𝑝(𝑥𝑥).

Common Core Algebra II Lesson 12: Graphing Factored Polynomials

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1. Consider the cubic polynomial function 𝑓𝑓(𝑥𝑥) = 𝑥𝑥3 − 𝑥𝑥2 − 12𝑥𝑥. Algebraically determine the zeros of this function. Then sketch a graph of this function.

Label the relative maximum and relative minimums of this function

2. Sketch a graph of a cubic polynomial with zeros at 𝑥𝑥 = 2, 𝑥𝑥 = 1, and 𝑥𝑥 − 3 on the set of axes below.

Write three different equations of polynomials with the zeros described above.


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3. Write the equation of the polynomial function graphed below.

4. Explain how the multiplicity of the zeros effect the graph of the polynomial.

5. Which graph represents 𝑓𝑓(𝑥𝑥) = (𝑥𝑥2 − 2𝑎𝑎𝑥𝑥 + 𝑎𝑎2)(𝑥𝑥 + 𝑏𝑏) where both 𝑎𝑎 > 0 and 𝑏𝑏 > 0?

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 12: Graphing Factored Polynomials

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Homework 1. State the zeros of the function given by 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 + 3)(𝑥𝑥 + 1)(𝑥𝑥 − 2). Graph 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) below.

2.Algebraically determine the zeros of the function 𝑓𝑓(𝑥𝑥) = 𝑥𝑥3 − 2𝑥𝑥2 − 𝑥𝑥 + 2. Sketch this function on the set of axes below.


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3. Let 𝑓𝑓(𝑥𝑥) = 𝑥𝑥4(𝑥𝑥 + 1)7(𝑥𝑥 − 1)2. State the zeros of 𝑓𝑓 and their multiplicity. Determine if 𝑓𝑓 passes through the 𝑥𝑥-axis or is tangent to the 𝑥𝑥-axis at each zero.

4. Create the equation of the cubic polynomial that has 𝑥𝑥-intercepts of 4,−3, and 2 that has a 𝑦𝑦-intercept of −18.

5. Sketch a graph of 𝑝𝑝(𝑥𝑥) = (𝑥𝑥 + 𝑎𝑎)(𝑥𝑥 − 𝑏𝑏)(𝑥𝑥 − 𝑐𝑐) if 𝑏𝑏 > 𝑐𝑐 on a set of axes below. Assume at 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 are positive.

Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 13: End Behavior of Polynomials

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Opening Exercise

A degree four polynomial with a leading coefficient of 1 is graphed below.

Write an equation for this polynomial, 𝑝𝑝(𝑥𝑥), in factored form.

1. The graphs of three polynomial functions are shown below. Describe the similarities in the graphs and equations of these functions.

𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 − 4𝑥𝑥 + 1 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥4 + 𝑥𝑥3 − 7𝑥𝑥2 − 𝑥𝑥 + 6 ℎ(𝑥𝑥) =12𝑥𝑥6 − 14𝑥𝑥4 + 49𝑥𝑥2 − 36

Common Core Algebra II Lesson 13: End Behavior of Polynomials

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3. Describe the end behavior of the polynomial function defined by the equation 𝑝𝑝(𝑥𝑥) = 3𝑥𝑥4 − 10𝑥𝑥3 + 𝑥𝑥2 − 𝑥𝑥 + 1.

4. Describe the end behavior of the polynomial function defined below. Is the leading coefficient positive or negative?

𝑓𝑓(𝑥𝑥) = −𝑥𝑥2 + 4𝑥𝑥 − 1 𝑔𝑔(𝑥𝑥) = −2𝑥𝑥4 − 𝑥𝑥3 + 7𝑥𝑥2 + 𝑥𝑥 − 6 ℎ(𝑥𝑥) = −12𝑥𝑥6 + 14𝑥𝑥4 − 49𝑥𝑥2 + 36


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𝑓𝑓(𝑥𝑥) = 𝑥𝑥3 + 4𝑥𝑥2 − 𝑥𝑥 − 4 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥5 − 𝑥𝑥4 − 7𝑥𝑥3 + 7𝑥𝑥2 + 16𝑥𝑥 + 16 ℎ(𝑥𝑥) =12𝑥𝑥7 + 4𝑥𝑥6 − 𝑥𝑥4 − 𝑥𝑥 + 4

𝑓𝑓(𝑥𝑥) = −𝑥𝑥3 + 4𝑥𝑥2 + 𝑥𝑥 − 4 𝑔𝑔(𝑥𝑥) = −2𝑥𝑥5 − 𝑥𝑥4 + 7𝑥𝑥3 + 7𝑥𝑥2 − 16𝑥𝑥 + 16 ℎ(𝑥𝑥) = −12𝑥𝑥7 + 4𝑥𝑥6 − 𝑥𝑥4 + 𝑥𝑥 + 4


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7. Let 𝑝𝑝(𝑥𝑥) = −3(𝑥𝑥 + 1)(𝑥𝑥 − 2)2(𝑥𝑥 + 4)5(𝑥𝑥 − 7)3.

What is the degree and leading coefficient of the 𝑝𝑝?

Describe the end behavior of 𝑝𝑝.

State the zeros of 𝑝𝑝 and their multiplicities. Use all this information to sketch 𝑝𝑝.

Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 13: End Behavior of Polynomials

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Homework 1. State the end behavior of the following functions without using a graphing calculator.

𝑓𝑓(𝑥𝑥) = −3𝑥𝑥4 + 5𝑥𝑥3 − 7𝑥𝑥2 + 𝑥𝑥 − 9 𝑔𝑔(𝑥𝑥) = 12𝑥𝑥5 + 3𝑥𝑥4 − 9𝑥𝑥2 − 3𝑥𝑥 + 1

2. Sketch a graph of the function 𝑓𝑓(𝑥𝑥) = 2(𝑥𝑥 + 1)(𝑥𝑥 − 3)(𝑥𝑥 + 5)2 by examining end behavior, the leading coefficient, and the zeros of the function.


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3. The following scatter plot represents the data from a study produced by the Center for Transportation Analysis. The study observed a car’s speed, in miles per hour, and also recorded the car’s fuel economy, in miles per gallon.

Determine if the polynomial function that models the data would have an even or odd degree. Is the leading coefficient of the polynomial that can be used to model this data positive or negative? Explain your reasoning.

4. Describe the end behavior of the graph of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) shown below. If 𝑓𝑓 is a polynomial, determine if the leading coefficient is positive or negative, and if the degree is even or odd?

5. Can you sketch a graph of an odd degree polynomial with no 𝑥𝑥-intercepts? Explain your reasoning.

Name:___________________________________________ Date:___________ Common Core Algebra II Lesson 14: Even and Odd Functions

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Opening Exercise

The function 𝑓𝑓(𝑥𝑥) is graphed below. Reflect 𝑓𝑓(𝑥𝑥) across the 𝑦𝑦-axis.

The function 𝑓𝑓(𝑥𝑥) is shown below. Rotate the function 180° about the origin.

1. If a function has symmetry across the 𝑦𝑦-axis, we call the function even.

If a function has symmetry across the origin, we call the function odd.

Common Core Algebra II Lesson 14: Even and Odd Functions

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2. Consider the function 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 4.

Find 𝑓𝑓(2) and 𝑓𝑓(−2). Find 𝑓𝑓(1) and 𝑓𝑓(−1).

Find 𝑓𝑓(−𝑥𝑥).

What do you notice? What does this tell us about 𝑓𝑓? Let’s look at the graph of 𝑓𝑓.

3. Consider the function 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥3 − 5𝑥𝑥.

Find 𝑔𝑔(2) and 𝑔𝑔(−2). Find 𝑔𝑔(1) and 𝑔𝑔(−1).

Find 𝑓𝑓(−𝑥𝑥)

What do you notice? What does this tell us about 𝑓𝑓? Let’s look at the graph of 𝑓𝑓.

Common Core Algebra II Lesson 14: Even and Odd Functions

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4. Are all functions either even or odd? Consider the function 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 − 4𝑥𝑥 + 1.

Find 𝑓𝑓(2) and 𝑓𝑓(−2). Find 𝑓𝑓(−𝑥𝑥).

What can we conclude about 𝑓𝑓?

5. For each graph shown, determine if the function is even, odd, or neither.

6. Describe the difference between the degree of a polynomial in terms of end behavior and even/odd functions.

Name:___________________________________________ Date:___________ Common Core Algebra II Lesson 14: Even and Odd Functions

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Homework 1. Half of the graph of 𝑓𝑓(𝑥𝑥) is shown below. Sketch the other half based on the function type.

2. Algebraically determine if 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥4 − 7𝑥𝑥2 + 5 is even, odd, or neither.

3. Describe each function as either even, odd, or neither.

Name:___________________________________________ Date:___________ Common Core Algebra II Lesson 15: Modeling with Polynomial Functions

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1. Your group will create a box from a piece of construction paper. Each group will record its box’s measurements and use said measurement values to calculate and record the volume of its box. Each group will contribute to the following class table on the board.

Using the given construction paper, cut out congruent squares from each corner, and fold the sides in order to create an open-topped box as shown on the figure.

Group Height (cm) Length Width Volume 1 2 3 4 5 6

What is the length and width of a box with a height of 𝑥𝑥?

Write a function 𝑉𝑉(𝑥𝑥) that represents the volume of a box with a height of 𝑥𝑥. Graph this function and determine an appropriate domain for this function.

What are the dimensions of the square that should be cut out to create the box with the largest volume?

Common Core Algebra II Lesson 15: Modeling with Polynomial Functions

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2. For a fundraiser, members of the math club decide to make and sell “Pythagoras may have been Fermat’s first problem but not his last!” t-shirts. They are trying to decide how many t-shirts to make and sell at a fixed price. They surveyed the level of interest of students around school and made a scatterplot of the number of t-shirts sold versus profit shown below.

a. Identify the 𝑦𝑦-intercept. Interpret its meaning within the context of this problem.

b. If we model this data with a function, what point on the graph of that function represents the number of t-shirts they need to sell in order to break even? Why?

c. What is the smallest number of t-shirts they can sell and still make a profit?

d. How many t-shirts should they sell in order to maximize the profit? What is the maximum profit?

e. Based on the scatterplot, would a quadratic or a cubic function be more appropriate to model this function? Explain your reasoning.

Common Core Algebra II Lesson 15: Modeling with Polynomial Functions

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Mr. Cerbino wants to find a function to model their data, so he drew cubic function through the data.

f. The function that models the profit in terms of the number of t-shirts made has the form 𝑃𝑃(𝑥𝑥) = 𝑐𝑐(𝑥𝑥3 − 53𝑥𝑥2 − 236𝑥𝑥 + 9828). Use the point labeled on the graph to find the value of the leading coefficient, 𝑐𝑐.

g. Find 𝑃𝑃(30) and interpret its meaning in the context of the problem?

h. Why do you think the function decreases?

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 16: What If There is a Remainder?

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Opening Exercise

Compute 1355 ÷ 4 using long division.

1. Find each quotient by inspection.

𝑥𝑥+4𝑥𝑥+1

𝑥𝑥2+4𝑥𝑥2+6

2𝑥𝑥−7𝑥𝑥−3

2. Find the following quotient using both reverse tabular method and long division.

𝑥𝑥3 + 2𝑥𝑥2 + 8𝑥𝑥 + 1𝑥𝑥 + 5

Common Core Algebra II Lesson 16: What If There is a Remainder?

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3. Find the following quotients using the method of your choice.

𝑥𝑥3−𝑥𝑥2+3𝑥𝑥−1𝑥𝑥+3

𝑥𝑥2+4𝑥𝑥+10𝑥𝑥−8

4𝑥𝑥3+5𝑥𝑥−82𝑥𝑥−5

2𝑥𝑥3−4𝑥𝑥2−7𝑥𝑥−10𝑥𝑥−5

4𝑥𝑥2−5𝑥𝑥2−1

𝑥𝑥4−8𝑥𝑥2+12𝑥𝑥+2

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 16: What If There is a Remainder?

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Homework 1. Create equivalent expression in the form of 𝑞𝑞(𝑥𝑥) + 𝑟𝑟(𝑥𝑥)

𝑏𝑏(𝑥𝑥) for each quotient.

𝑥𝑥3+7𝑥𝑥2+14𝑥𝑥+3𝑥𝑥+2

2𝑥𝑥2−13𝑥𝑥−10

2𝑥𝑥+3

𝑥𝑥−2𝑥𝑥+1

9𝑥𝑥3−12𝑥𝑥2+4

𝑥𝑥−2

2. Let 𝑃𝑃(𝑥𝑥) = 𝑥𝑥3 + 2𝑥𝑥2 + 2𝑥𝑥 − 5. Divide 𝑃𝑃 by 𝑥𝑥 − 1. Then find 𝑃𝑃(1). Explain what you observe.

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 17: The Remainder Theorem

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Opening Exploration

Consider the polynomial function 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥2 + 8𝑥𝑥 − 4.

a. Divide 𝑓𝑓 by 𝑥𝑥 − 2. b. Find 𝑓𝑓(2).

Consider the polynomial function 𝑔𝑔(𝑥𝑥) = 𝑥𝑥3 − 3𝑥𝑥2 + 6𝑥𝑥 + 8.

a. Divide 𝑔𝑔 by 𝑥𝑥 + 1. b. Find 𝑔𝑔(−1).

Consider the polynomial ℎ(𝑥𝑥) = 𝑥𝑥3 + 2𝑥𝑥 − 3.

a. Divide ℎ by 𝑥𝑥 − 3. b. Find ℎ(3).

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The Remainder Theorem:

1. Determine if 𝑥𝑥 − 4 is a factor of the function 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥3 − 5𝑥𝑥2 − 11𝑥𝑥 − 4 using two different methods.

2. Determine if 𝑥𝑥 + 2 is a factor of the function 𝑝𝑝(𝑥𝑥) = 𝑥𝑥3 + 𝑥𝑥2 − 27𝑥𝑥 − 15 using two different methods.

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3. Consider the polynomial 𝑃𝑃(𝑥𝑥) + 𝑥𝑥3 + 𝑘𝑘𝑥𝑥2 + 𝑥𝑥 + 6. Find the value of 𝑘𝑘 so that 𝑥𝑥 + 1 is a factor of 𝑃𝑃.

4. Find 𝑔𝑔(−1) if 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥3 − 3𝑥𝑥2 − 17𝑥𝑥 − 12 and explain what your answer tells you about 𝑥𝑥 + 1 as a factor of 𝑔𝑔.

5. Let 𝑦𝑦 = 𝑝𝑝(𝑥𝑥) be the graph shown below. Find the remainder when 𝑝𝑝 is divided by 𝑥𝑥 − 1. Explain your reasoning.

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 17: The Remainder Theorem

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Homework 1. Let 𝑓𝑓(𝑥𝑥) = 𝑥𝑥3 − 6𝑥𝑥2 − 7𝑥𝑥 + 9. Divide 𝑓𝑓 by 𝑥𝑥 − 3. Check your work by finding 𝑓𝑓(3).

2. Is 𝑥𝑥 + 1 a factor of 2𝑥𝑥5 − 4𝑥𝑥4 + 9𝑥𝑥3 − 𝑥𝑥 + 13? Explain your reasoning.

3. Which of the following binomials is a factor of 𝑥𝑥3 + 3𝑥𝑥2 − 10𝑥𝑥 − 24.

(1) 𝑥𝑥 − 1 (3) 𝑥𝑥 − 3

(2) 𝑥𝑥 − 2 (4) 𝑥𝑥 − 4

4. The function 𝑓𝑓 is defined by a polynomial. Some values of 𝑥𝑥 and 𝑓𝑓(𝑥𝑥) are shown in the table below. What is the remainder when 𝑓𝑓 is divide by 𝑥𝑥 + 2? Explain your reasoning.

𝑥𝑥 𝑓𝑓(𝑥𝑥) −3 −100 −2 −42 −1 −9 0 5 1 6 2 0 3 −7 4 −9

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 18: Putting It All Together

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Opening Exploration

Verify that 𝑥𝑥 + 1 is a factor of 𝑃𝑃(𝑥𝑥) = 2𝑥𝑥3 + 3𝑥𝑥2 − 2𝑥𝑥 − 3. Explain your reasoning.

1. Knowing that 𝑥𝑥 + 1 is factor of 𝑃𝑃, write 𝑃𝑃(𝑥𝑥) as the product of three linear factors.

State the zeros of 𝑃𝑃.

Sketch a graph of 𝑦𝑦 = 𝑃𝑃(𝑥𝑥).

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2. The polynomial function 𝑝𝑝(𝑥𝑥) = 2𝑥𝑥4 − 3𝑥𝑥3 − 15𝑥𝑥2 + 31𝑥𝑥 − 15 has a zero at 𝑥𝑥 = 1 of multiplicity of 2.

Find all real zeros of 𝑝𝑝.

Describe the end behavior of 𝑝𝑝.

Sketch a graph of 𝑝𝑝 showing the end behavior and zeros of 𝑝𝑝.

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3. Given 𝑧𝑧(𝑥𝑥) = 6𝑥𝑥3 + 𝑏𝑏𝑥𝑥2 − 52𝑥𝑥 + 15, 𝑧𝑧(2) = 35, and 𝑧𝑧(−5) = 0, algebraically determine the zeros of 𝑧𝑧(𝑥𝑥).

4. Consider the polynomial 𝑃𝑃(𝑥𝑥) = 𝑥𝑥3 + 𝑘𝑘𝑥𝑥2 − 10𝑥𝑥 + 24. Find the value of 𝑘𝑘 so that 𝑥𝑥 − 4 is a factor of 𝑃𝑃.

Express 𝑃𝑃 as the product of three linear factors.

Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 18: Putting It All Together

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Homework 1. Let 𝑝𝑝(𝑥𝑥) = 2𝑥𝑥4 + 11𝑥𝑥3 − 3𝑥𝑥2 − 44𝑥𝑥 − 20.

Explain why 𝑥𝑥 + 5 is a factor of 𝑝𝑝.

Knowing 𝑥𝑥 + 5 is a factor, factor 𝑝𝑝 completely.

State the solutions to 𝑝𝑝(𝑥𝑥) = 0.

Describe the end behavior of 𝑝𝑝 and sketch graph.

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Unit 1- Polynomial Functions