Unit 1- Polynomial Functions
76
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
Transcript of 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 β’ 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.
β’ A.APR.B.2- Know and apply the Remainder Theorem: For a polynomial ππ(π₯π₯) and a number ππ, the remainder on division by π₯π₯ β ππ is ππ(ππ), so ππ(ππ) = 0 if and only if π₯π₯ β ππ is a factor of ππ(π₯π₯).
β’ 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.
β’ A.APR.C.4- Prove polynomial identities and use them to describe numerical
relationships.
β’ A.APR.D.6- Rewrite simple rational expressions in different forms; write ππ(π₯π₯)/ππ(π₯π₯) in the form ππ(π₯π₯) + ππ(π₯π₯)/ππ(π₯π₯), where ππ(π₯π₯), ππ(π₯π₯), ππ(π₯π₯), and ππ(π₯π₯) are polynomials with the degree of ππ(π₯π₯) less than the degree of ππ(π₯π₯), using inspection or long division.
β’ F.IF.C.7c- Graph polynomial functions, identifying zeros when suitable
factorizations are available, and showing end behavior.
β’ 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 Γ 27 = (20 + 8)(20 + 7) using an area model.
1. Use the tabular method to multiply (π₯π₯ + 8)(π₯π₯ + 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
2
2. Use the tabular method to multiply (π₯π₯2 + 3π₯π₯ + 1)(π₯π₯2 β 5π₯π₯ + 2) and combine like terms.
3. Use the tabular method to multiply (π₯π₯2 + 3π₯π₯ + 1)(π₯π₯2 β 2) and combine like terms.
4. Using the distributive property, express the product (π₯π₯ β 1)(π₯π₯2 + π₯π₯ + 1) in standard form.
Common Core Algebra II Lesson 1: The Multiplication of Polynomials
3
5. Find the product (π₯π₯ β 1)(π₯π₯3 + π₯π₯2 + π₯π₯ + 1) using table.
6. Using exercises 3 & 4, generalize the pattern that emerges by writing an identity for (π₯π₯ β 1)(π₯π₯ππ + π₯π₯ππβ1 + β―+ π₯π₯2 + π₯π₯ + 1) for positive integer ππ.
7. Create an equivalent expression to (ππ + ππ + ππ)2.
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 1: The Multiplication of Polynomials
4
Homework Multiply the following polynomials and express your answer in standard form.
1. (π₯π₯2 β 4π₯π₯ + 4)(π₯π₯ + 3) 2. (2π₯π₯ β 3)(π₯π₯3 + π₯π₯2 + π₯π₯ + 1)
3. (π₯π₯2 β 3π₯π₯ + 9)(π₯π₯2 + 3π₯π₯ + 9) 4. (π‘π‘ + 1)(π‘π‘ β 1)(π‘π‘2 + 1)
5. If ππ(π₯π₯) = 2π₯π₯ + 1 and ππ(π₯π₯) = 2π₯π₯2 + 1, express ππ(π₯π₯) β ππ(π₯π₯) 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.
(2π₯π₯ + 5)(π₯π₯2 + 5π₯π₯ + 1)
How can you use your answer from above to quickly multiply 25 β 151?
1. Show that 2π₯π₯3+15π₯π₯2+27π₯π₯+5
2π₯π₯+5 = π₯π₯2 + 5π₯π₯ + 1
Common Core Algebra II Lesson 2: The Division of Polynomials
6
2. Reverse the tabular of multiplication to find the quotient 2π₯π₯2+π₯π₯β10
π₯π₯β2.
3. Create your own table and use the ππππππππππππππ π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ πππππ‘π‘βππππ to find the quotient.
π₯π₯4 + 4π₯π₯3 + 3π₯π₯2 + 4π₯π₯ + 2π₯π₯2 + 1
Explain how we can use the previous quotient to factor π₯π₯4 + 4π₯π₯3 + 3π₯π₯2 + 4π₯π₯ + 2.
Common Core Algebra II Lesson 2: The Division of Polynomials
7
4. Use the ππππππππππππππ π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ πππππ‘π‘βππππ to find the quotient.
3π₯π₯5 β 2π₯π₯4 + 6π₯π₯3 β 4π₯π₯2 β 24π₯π₯ + 16π₯π₯2 + 4
OYO!
5. Find the following quotients:
4π₯π₯3β10π₯π₯2β22π₯π₯β82π₯π₯+1
π₯π₯4+3π₯π₯3β6π₯π₯2β6π₯π₯+8π₯π₯+4
Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 2: The Division of Polynomials
8
Homework 1. Use the reverse tabular method to find the following quotients.
2π₯π₯3+π₯π₯2β16π₯π₯+152π₯π₯β3
π₯π₯3β8π₯π₯β2
3π₯π₯5+12π₯π₯4+11π₯π₯3+2π₯π₯2β4π₯π₯β23π₯π₯2β1
4π₯π₯2+8π₯π₯+32π₯π₯+1
2. First compute 3π₯π₯3+10π₯π₯2β14π₯π₯+4
3π₯π₯β2. Then express 3π₯π₯3 + 10π₯π₯2 β 14π₯π₯ + 4 as the product of
two polynomials. Explain your reasoning.
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 3: Long Division, Again?
9
Opening Exercise
Use the reverse tabular method to determine the quotient 2π₯π₯3+11π₯π₯2+7π₯π₯+10
π₯π₯+5.
Write the polynomial 2π₯π₯3 + 11π₯π₯2 + 7π₯π₯ + 10 in factored form.
1. Take a trip back to elementary school and use long division to evaluate 1573 Γ· 13.
Common Core Algebra II Lesson 3: Long Division, Again?
10
2. Letβs return to back to Algebra 2. If we let π₯π₯ = 10, then we can represent the previous problem as
3753 23 ++++ xxxx
3. Use the long division algorithm for polynomials to evaluate 2π₯π₯3 β 4π₯π₯2 + 2
2π₯π₯ β 2.
Common Core Algebra II Lesson 3: Long Division, Again?
11
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. 7π₯π₯3β8π₯π₯2β13π₯π₯+2
7π₯π₯β1 b. π₯π₯
2+6π₯π₯+9π₯π₯+3
c. π₯π₯3β27π₯π₯β3
d. 2π₯π₯4+14π₯π₯3+π₯π₯2β21π₯π₯β6
2π₯π₯2β3
Common Core Algebra II Lesson 3: Long Division, Again?
12
e. 5π₯π₯4β6π₯π₯2+1
π₯π₯2β1 f.
π₯π₯3+2π₯π₯2+4π₯π₯+8π₯π₯+2
g. 2π₯π₯7+π₯π₯5β4π₯π₯3+14π₯π₯2β2π₯π₯+7
2π₯π₯2+1 h.
π₯π₯6β64π₯π₯+2
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 3: Long Division, Again?
13
Homework 1. Use the long division algorithm to determine the quotients.
2π₯π₯3β13π₯π₯2βπ₯π₯+32π₯π₯+1
3π₯π₯3+4π₯π₯2+7π₯π₯+22
π₯π₯+2
2. Given ππ(π₯π₯) = 4π₯π₯3 + 5π₯π₯ + 21 and β(π₯π₯) = 2π₯π₯ + 3, express ππ(π₯π₯)β(π₯π₯)
in standard form by using
long division.
3. Given ππ(π₯π₯) = 3π₯π₯3 β 4π₯π₯2 + 5π₯π₯ + ππ, determine the value of ππ so that 3π₯π₯ β 7 is a factor of the polynomial ππ. Explain your reasoning.
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 4: Operations with Polynomials
14
Opening Exercise
Find the sum of 3π₯π₯2 β 7π₯π₯ + 11 and 5π₯π₯3 β 22π₯π₯2 + 9π₯π₯ β 5.
Find the difference 3π₯π₯2 β 7π₯π₯ + 11 β (5π₯π₯3 β 22π₯π₯2 + 9π₯π₯ β 5).
1. We can combine operations together. Rewrite each polynomial in standard form by applying the operations in the appropriate order.
a. (π₯π₯2+5π₯π₯+20)+(π₯π₯2+6π₯π₯β6)
π₯π₯+2 b. (π₯π₯2 β 4)(π₯π₯ + 2) β 3(π₯π₯2 + 2π₯π₯ β 5)
2. A manufacture has developed a cost model, πΆπΆ(π₯π₯) = 0.15π₯π₯3 + 0.01π₯π₯2 + 2π₯π₯ + 120, where π₯π₯ is the number of items sold, in thousands. The sales price can be modeled by ππ(π₯π₯) = 30 β 0.01π₯π₯. Therefore, revenue is modeled by π π (π₯π₯) = π₯π₯ β ππ(π₯π₯).
Write a polynomial in standard form that can be used to model the companyβs profits ππ(π₯π₯) = π π (π₯π₯) β πΆπΆ(π₯π₯).
Common Core Algebra II Lesson 4: Operations with Polynomials
15
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
16
More space to work!
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 4: Operations with Polynomials
17
Homework 1. Perform the indicated operations to write each polynomial in standard form.
(2π₯π₯2 β π₯π₯3 β 9π₯π₯ + 1) β 4(π₯π₯3 + 7π₯π₯ β 3π₯π₯2 + 1) (π₯π₯ + 3)2 β (π₯π₯ + 4)2
π₯π₯2β5π₯π₯+6π₯π₯β3
+π₯π₯2 + π₯π₯ + 1 (π₯π₯ + 3)(π₯π₯ β 3) β (π₯π₯ + 4)(π₯π₯ β 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
18
Opening Exercise
Show that the sum of three consecutive integers is the three times the middle integer.
1. Prove that if π₯π₯ > 1, then a triangle with side lengths π₯π₯2 β 1, 2π₯π₯, and π₯π₯2 + 1 is a right triangle.
Pick a value of π₯π₯ to create a Pythagorean triple.
Explain why every Pythagorean triple must contain an even integer.
Common Core Algebra II Lesson 5: Polynomial Identities
19
2. Prove that (π₯π₯ + ππ)2 = π₯π₯2 + 2πππ₯π₯ + ππ2 is an identity.
3. Use this identity to quickly compute the following expressions.
(π₯π₯ + 5)2 (3π¦π¦ β 4)2 (5π₯π₯ + 2π¦π¦)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
20
5. Algebraically determine the values of ππ and ππ to correctly complete the identity stated below.
4π₯π₯3 + πππ₯π₯2 + 23π₯π₯ + 20 = (2π₯π₯ + 5)(2π₯π₯2 + πππ₯π₯ + 4)
6. Prove the identity (ππ2 + ππ2)(π₯π₯2 + π¦π¦2) = (πππ₯π₯ β πππ¦π¦)2 + (πππ₯π₯ + πππ¦π¦)2.
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 5: Polynomial Identities
21
Homework 1. Prove that (π₯π₯ + π¦π¦)2 β (π₯π₯ β π¦π¦)2 = 4π₯π₯π¦π¦ for all real numbers π₯π₯ and π¦π¦.
2. Prove that (ππ + ππ)3 = ππ3 + 3ππ2ππ + 3ππππ2 + ππ3 is an identity.
3. The identity (π₯π₯2 + π¦π¦2)2 = (π₯π₯2 β π¦π¦2)2 + (2π₯π₯π¦π¦)2 can be used to generate Pythagorean triples. Show that this statement is an identity.
Pick values for π₯π₯ and π¦π¦, where π₯π₯ > π¦π¦, to generate a Pythagorean triple.
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 6: GCF and The Difference of Squares
22
Opening Exercise
Prove the polynomial identity ππ2 β ππ2 = (ππ + ππ)(ππ β ππ).
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.
6π₯π₯2 + 18π₯π₯ + 10 3π₯π₯3 + 18π₯π₯
2ππ4 + 6ππ3 β 18ππ2 β 54ππ 5π₯π₯3π¦π¦ β 30π₯π₯2π¦π¦2 β 3π₯π₯π¦π¦3
Sometimes the GCF can be more than just monomial!
2. Factor out the GCF from each of the following expression.
2π₯π₯(π₯π₯ + 5) β 3(π₯π₯ + 5) π₯π₯2(3π₯π₯ + 5) + 16(3π₯π₯ + 5)
π‘π‘(π‘π‘2 + 5π‘π‘ + 6) β 2(π‘π‘2 + 5π‘π‘ + 6) ππ2(ππ + 4) + 8ππ(ππ + 4) + 12(ππ + 4)
Common Core Algebra II Lesson 6: GCF and The Difference of Squares
23
We also factored the difference of squares in Algebra I.
3. Factor the following expression completely.
π₯π₯2 β 25 1 β 9π¦π¦2 81π¦π¦4 β 16π₯π₯4
What about the sum of perfect squares? Itβs reasonable to think that π₯π₯ + ππ or π₯π₯ β ππ could be a factor of π₯π₯2 + ππ2.
4. Compute the quotients below if possible.
π₯π₯2 + 4π₯π₯ + 2
π₯π₯2 + 4π₯π₯ β 2
5. Factor the expressions completely.
π¦π¦ β π¦π¦5 2π₯π₯3 β 18π₯π₯ π₯π₯8 β 1
Common Core Algebra II Lesson 6: GCF and The Difference of Squares
24
6. The expression π₯π₯2(π₯π₯ β π¦π¦)3 β π¦π¦2(π₯π₯ β π¦π¦)3 can be rewritten as (π₯π₯ + π¦π¦)(π₯π₯ β π¦π¦)ππ. Determine and state the value of ππ.
7. Factor the expression (π₯π₯ β 1)2 β 4 as the difference of perfect squares.
8. Factor the expression ππ2(π₯π₯4 β π¦π¦4) β 4(π₯π₯4 β π¦π¦4) completely.
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 6: GCF and The Difference of Squares
25
Homework 1. Factor the expressions completely.
π₯π₯4 β 81 9ππ2 β 49
2. Factor, over the integers, the expression 12π‘π‘8 β 75π‘π‘4 completely.
3. Factor the expressions completely.
4π₯π₯2(π₯π₯ + 5) β 9(π₯π₯ + 5) π₯π₯2(6π₯π₯ β 5π¦π¦) β 4π¦π¦2(6π₯π₯ β 5π¦π¦)
4. Explain why the expression π₯π₯2 + 1 cannot be factored over the real numbers as (π₯π₯ + 1)(π₯π₯ β 1).
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 7: Perfect Cubes
26
Opening Exercise
Write out the list of perfect cubes from 1 to 6.
Explain why we call π₯π₯3 β ππ3 the difference of perfect cubes.
1. Find the quotient of π₯π₯3βππ3
π₯π₯βππ. Explain how you can use this to factor π₯π₯3 β ππ3.
2. Factor the expression completely.
π₯π₯3 β 27 125π¦π¦3 β 1 2π§π§4 β 16π§π§
Common Core Algebra II Lesson 7: Perfect Cubes
27
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 π₯π₯3+ππ3
π₯π₯+ππ. Explain how you can use this to factor π₯π₯3 + ππ3.
4. Factor the following expressions completely.
ππ3 + 216 2π₯π₯5 + 128π₯π₯2 27π₯π₯3 + 8π§π§3
5. Factor the expression π₯π₯6 β 1 completely.
Common Core Algebra II Lesson 7: Perfect Cubes
28
OYO
6. Factor each of the following expressions completely.
π₯π₯3 β 125 π₯π₯3 β 216π¦π¦3
8π₯π₯4 + π₯π₯ π₯π₯3(π₯π₯ + 4) + 64(π₯π₯ + 4)
ππ3 β 8ππ3 128π₯π₯4 + 54π₯π₯π¦π¦3
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 7: Perfect Cubes
29
Homework Factor the expressions completely.
π₯π₯3 + 8 1 β π₯π₯3
792π₯π₯6 + 64π¦π¦6 π₯π₯3 β π₯π₯
125π§π§3 + 1 ππ4 β ππ4
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 8: Factoring Trinomial Review
30
Opening Exercise
Find the product of (π₯π₯ + 6)(π₯π₯ + 3).
Factor the following polynomials completely.
π₯π₯2 + 8π₯π₯ + 15 3π₯π₯2 + 12π₯π₯ β 15 βπ₯π₯3 + 12π₯π₯2 β 20π₯π₯
The ππ β ππ Method
Factor the trinomial 2π₯π₯2 + 11π₯π₯ + 12.
Common Core Algebra II Lesson 7: Factoring Trinomial Review
31
Use the ππ β ππ method to factor the trinomials below.
a. 3π₯π₯2 + 17π₯π₯ β 6 b. 4π₯π₯2 + 4π₯π₯ β 15
c. 6π₯π₯2 + 7π₯π₯ β 20 d. β3π₯π₯2 β 5π₯π₯ + 2
e. 6π₯π₯2 β 11π₯π₯ + 3 f. 16π₯π₯2 β 8π₯π₯ β 3
Name:__________________________________________________ Date:___________ Common Core Algebra II Lesson 8: Factoring Trinomial Review
32
Homework Factoring the following polynomials completely.
6π₯π₯2 + 48π₯π₯ + 90 3π₯π₯2 + 4π₯π₯ β 20
4π‘π‘2 + 25π‘π‘ + 25 5π₯π₯3 β 41π₯π₯2 + 8π₯π₯
8ππ2 + 20ππ β 12 9π₯π₯4 + 35π₯π₯2 β 4
Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 9: Seeing Structure: Grouping
33
Opening Exercise
Factor the trinomial 8π₯π₯2 β 10π₯π₯ + 3.
1. Factor the expression below completely by grouping the terms.
π₯π₯3 β 5π₯π₯2 β 4π₯π₯ + 20
2. Use grouping to factor the cubic polynomials below completely if it can be factored.
π₯π₯3 β 8π₯π₯ + 2π₯π₯ β 16 π₯π₯3 + 2π₯π₯2 β π₯π₯ + 2 4π₯π₯3 + 2π₯π₯2 β 36π₯π₯ β 18
Common Core Algebra II Lesson 9: Seeing Structure: Grouping
34
3. This polynomial isnβt cubic, but maybe we can still use grouping to factor it completely.
π₯π₯4 + 5π₯π₯3 + 8π₯π₯ + 40
4. The concept of βgroupingβ can be used to factor some interesting expressions.
ππ4 β 4ππ2 + 5ππ3 β 20ππ + 6ππ2 β 24
π₯π₯5 + 5π₯π₯4 + 4π₯π₯3 + π₯π₯2 + 5π₯π₯ + 4
Common Core Algebra II Lesson 9: Seeing Structure: Grouping
35
5. Express 6ππ3β30ππ2+4ππ2β20ππβ2ππ+10
ππ+1 as the product of linear factors.
We have learned a variety of factoring strategies this in this unit. Letβs organize our techniques of factoring polynomials.
Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 9: Seeing Structure: Grouping
36
Homework 1. Factor the following polynomials completely.
π₯π₯3 β 6π₯π₯2 β 25π₯π₯ + 150 3π₯π₯3 β 5π₯π₯2 β 48π₯π₯ + 80
π₯π₯4 β 4π₯π₯3 + 4π₯π₯2 β 16π₯π₯ 4π₯π₯3 + 2π₯π₯2 β 36π₯π₯ β 18
2. Explain why π₯π₯3 + 3π₯π₯2 β 2π₯π₯ + 6 cannot be factored using grouping.
Common Core Algebra II Lesson 9: Seeing Structure: Grouping
37
3. Factor the expressions completely.
6π₯π₯3 β 5π₯π₯2π¦π¦ β 24π₯π₯π¦π¦2 + 20π¦π¦3
π₯π₯3 β 3π₯π₯2 β 6π₯π₯2 + 18π₯π₯ + 8π₯π₯ β 24
2π₯π₯4 β 7π₯π₯3 β 15π₯π₯2 β 8π₯π₯2 + 28π₯π₯ + 60
Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 10: Seeing Structure: Advance Factoring
38
Opening Exercise
Factor the polynomial 4π₯π₯5 β π₯π₯4 β 4π₯π₯3 + π₯π₯2 completely.
Todayβs main goal is to βLook for and make use of structureβ 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.
1. Factor the expressions completely.
a. π₯π₯2 + 4π₯π₯ + 3 b. (2π₯π₯ + 1)2 + 4(2π₯π₯ + 1) + 3
c. π₯π₯2 β 13π₯π₯ + 36 d. π₯π₯4 β 13π₯π₯2 + 36
Common Core Algebra II Lesson 10: Seeing Structure: Advance Factoring
39
e. π₯π₯2 + 5π₯π₯ β 6 f. (π₯π₯3 + 2)2 + 5(π₯π₯3 + 2) β 6
OYO!
As usual, factor completely.
(π₯π₯2 + 4π₯π₯)2 β 16 π¦π¦6 β 7π¦π¦3 β 8
(3π₯π₯2 β π₯π₯)2 β 32(3π₯π₯2 β π₯π₯) + 60
Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 10: Seeing Structure: Advance Factoring
40
Homework Factor the following polynomials completely.
1. π¦π¦4 β 21π¦π¦2 β 100 2. ππ6 β ππ4 β 16ππ2 + 16
3. (2π₯π₯2 β 7π₯π₯)2 β (2π₯π₯2 β 7π₯π₯) β 12 4. 8π₯π₯6 + 7π₯π₯3 β 1
Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 11: The Special Role of Zero in Factoring
41
Opening Exercise
For each equation, list some possible values of π₯π₯ and π¦π¦.
π₯π₯π¦π¦ = 10 π₯π₯π¦π¦ = 1 π₯π₯π¦π¦ = β1 π₯π₯π¦π¦ = 0
Does one equation tell you more information than the others?
The Zero Product Property:
1. Find all solutions to the equation (π₯π₯2 + 5π₯π₯ + 6)(π₯π₯2 β 3π₯π₯ β 4) = 0.
2. Find all solutions to the equation (π₯π₯3 β 9π₯π₯)(π₯π₯3 + π₯π₯2 β π₯π₯ β 1) = 0
Common Core Algebra II Lesson 11: The Special Role of Zero in Factoring
42
3. Suppose we know that the polynomial equation 2π₯π₯3 + 9π₯π₯2 + π₯π₯ β 12 = 0 has three real solutions and that one of the factors of 2π₯π₯3 + 9π₯π₯2 + π₯π₯ β 12 is π₯π₯ β 1. How can we find all three solutions to the given equation?
Letβs look at this the polynomial ππ(π₯π₯) = 2π₯π₯3 + 9π₯π₯2 + π₯π₯ β 12 graphically.
Factor-Zero Theorem:
Common Core Algebra II Lesson 11: The Special Role of Zero in Factoring
43
4. Consider the polynomial functions ππ(π₯π₯) = (π₯π₯ β 2)(π₯π₯ + 3)2, ππ(π₯π₯) = (π₯π₯ β 2)2(π₯π₯ + 3)4, and ππ(π₯π₯) = (π₯π₯ β 2)4(π₯π₯ β 3)5.
Quickly, find the zeros of all three functions.
5. Find the zeros of the following polynomial functions, with their multiplicities.
a. ππ(π₯π₯) = (π₯π₯ + 1)(π₯π₯ β 1)(π₯π₯2 + 1) b. ππ(π₯π₯) = (π₯π₯ β 4)3(π₯π₯ β 2)8
c. β(π₯π₯) = (2π₯π₯ β 3)5 d. ππ(π₯π₯) = (3π₯π₯ + 4)100(π₯π₯ β 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
44
Homework
1. Find all real solutions to the given equations.
(π₯π₯ β 5)(3π₯π₯ + 2)(π₯π₯ + 3) = 0 (4π₯π₯2 β 9)(π₯π₯2 β 16) = 0
6π₯π₯3 β 27π₯π₯2 β 15π₯π₯ = 0 π₯π₯3 + 3π₯π₯2 β 4π₯π₯ = 12
Common Core Algebra II Lesson 11: The Special Role of Zero in Factoring
45
2. Determine all real zeros and state their multiplicity for the function below.
ππ(π₯π₯) = (π₯π₯2 β 16)(π₯π₯3 + 4π₯π₯2 β 16π₯π₯ β 64)
3. Suppose we know that π₯π₯ + 2 is a factor of ππ(π₯π₯) = 4π₯π₯3 + 12π₯π₯2 + 5π₯π₯ β 6. Find the other factors of ππ and use them to determine when ππ(π₯π₯) = 0.
Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 12: Graphing Factored Polynomials
46
Opening Exercise
Find algebraically the zeros of ππ(π₯π₯) = π₯π₯3 β π₯π₯2 β 4π₯π₯ + 4.
On the set of axes below, graph π¦π¦ = ππ(π₯π₯).
Explain what the zeros represent on the graph of π¦π¦ = ππ(π₯π₯).
Common Core Algebra II Lesson 12: Graphing Factored Polynomials
47
1. Consider the cubic polynomial function ππ(π₯π₯) = π₯π₯3 β π₯π₯2 β 12π₯π₯. 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 π₯π₯ = 2, π₯π₯ = 1, and π₯π₯ β 3 on the set of axes below.
Write three different equations of polynomials with the zeros described above.
Common Core Algebra II Lesson 12: Graphing Factored Polynomials
48
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 ππ(π₯π₯) = (π₯π₯2 β 2πππ₯π₯ + ππ2)(π₯π₯ + ππ) where both ππ > 0 and ππ > 0?
Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 12: Graphing Factored Polynomials
49
Homework 1. State the zeros of the function given by ππ(π₯π₯) = (π₯π₯ + 3)(π₯π₯ + 1)(π₯π₯ β 2). Graph π¦π¦ = ππ(π₯π₯) below.
2.Algebraically determine the zeros of the function ππ(π₯π₯) = π₯π₯3 β 2π₯π₯2 β π₯π₯ + 2. Sketch this function on the set of axes below.
Common Core Algebra II Lesson 12: Graphing Factored Polynomials
50
3. Let ππ(π₯π₯) = π₯π₯4(π₯π₯ + 1)7(π₯π₯ β 1)2. State the zeros of ππ and their multiplicity. Determine if ππ passes through the π₯π₯-axis or is tangent to the π₯π₯-axis at each zero.
4. Create the equation of the cubic polynomial that has π₯π₯-intercepts of 4,β3, and 2 that has a π¦π¦-intercept of β18.
5. Sketch a graph of ππ(π₯π₯) = (π₯π₯ + ππ)(π₯π₯ β ππ)(π₯π₯ β ππ) if ππ > ππ on a set of axes below. Assume at ππ, ππ, and ππ are positive.
Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 13: End Behavior of Polynomials
51
Opening Exercise
A degree four polynomial with a leading coefficient of 1 is graphed below.
Write an equation for this polynomial, ππ(π₯π₯), in factored form.
1. The graphs of three polynomial functions are shown below. Describe the similarities in the graphs and equations of these functions.
ππ(π₯π₯) = π₯π₯2 β 4π₯π₯ + 1 ππ(π₯π₯) = 2π₯π₯4 + π₯π₯3 β 7π₯π₯2 β π₯π₯ + 6 β(π₯π₯) =12π₯π₯6 β 14π₯π₯4 + 49π₯π₯2 β 36
Common Core Algebra II Lesson 13: End Behavior of Polynomials
52
2. The graphs of three polynomial functions are shown below. Describe the similarities in the graphs and equations of these functions.
3. Describe the end behavior of the polynomial function defined by the equation ππ(π₯π₯) = 3π₯π₯4 β 10π₯π₯3 + π₯π₯2 β π₯π₯ + 1.
4. Describe the end behavior of the polynomial function defined below. Is the leading coefficient positive or negative?
ππ(π₯π₯) = βπ₯π₯2 + 4π₯π₯ β 1 ππ(π₯π₯) = β2π₯π₯4 β π₯π₯3 + 7π₯π₯2 + π₯π₯ β 6 β(π₯π₯) = β12π₯π₯6 + 14π₯π₯4 β 49π₯π₯2 + 36
Common Core Algebra II Lesson 13: End Behavior of Polynomials
53
5. The graphs of three polynomial functions are shown below. Describe the similarities in the graphs and equations of these functions.
6. The graphs of three polynomial functions are shown below. Describe the similarities in the graphs and equations of these functions.
ππ(π₯π₯) = π₯π₯3 + 4π₯π₯2 β π₯π₯ β 4 ππ(π₯π₯) = 2π₯π₯5 β π₯π₯4 β 7π₯π₯3 + 7π₯π₯2 + 16π₯π₯ + 16 β(π₯π₯) =12π₯π₯7 + 4π₯π₯6 β π₯π₯4 β π₯π₯ + 4
ππ(π₯π₯) = βπ₯π₯3 + 4π₯π₯2 + π₯π₯ β 4 ππ(π₯π₯) = β2π₯π₯5 β π₯π₯4 + 7π₯π₯3 + 7π₯π₯2 β 16π₯π₯ + 16 β(π₯π₯) = β12π₯π₯7 + 4π₯π₯6 β π₯π₯4 + π₯π₯ + 4
Common Core Algebra II Lesson 13: End Behavior of Polynomials
54
7. Let ππ(π₯π₯) = β3(π₯π₯ + 1)(π₯π₯ β 2)2(π₯π₯ + 4)5(π₯π₯ β 7)3.
What is the degree and leading coefficient of the ππ?
Describe the end behavior of ππ.
State the zeros of ππ and their multiplicities. Use all this information to sketch ππ.
Name:____________________________________________ Date:___________ Common Core Algebra II Lesson 13: End Behavior of Polynomials
55
Homework 1. State the end behavior of the following functions without using a graphing calculator.
ππ(π₯π₯) = β3π₯π₯4 + 5π₯π₯3 β 7π₯π₯2 + π₯π₯ β 9 ππ(π₯π₯) = 12π₯π₯5 + 3π₯π₯4 β 9π₯π₯2 β 3π₯π₯ + 1
2. Sketch a graph of the function ππ(π₯π₯) = 2(π₯π₯ + 1)(π₯π₯ β 3)(π₯π₯ + 5)2 by examining end behavior, the leading coefficient, and the zeros of the function.
Common Core Algebra II Lesson 13: End Behavior of Polynomials
56
3. The following scatter plot represents the data from a study produced by the Center for Transportation Analysis. The study observed a carβs speed, in miles per hour, and also recorded the carβs 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 π¦π¦ = ππ(π₯π₯) shown below. If ππ 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 π₯π₯-intercepts? Explain your reasoning.
Name:___________________________________________ Date:___________ Common Core Algebra II Lesson 14: Even and Odd Functions
57
Opening Exercise
The function ππ(π₯π₯) is graphed below. Reflect ππ(π₯π₯) across the π¦π¦-axis.
The function ππ(π₯π₯) is shown below. Rotate the function 180Β° about the origin.
1. If a function has symmetry across the π¦π¦-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
58
2. Consider the function ππ(π₯π₯) = π₯π₯2 + 4.
Find ππ(2) and ππ(β2). Find ππ(1) and ππ(β1).
Find ππ(βπ₯π₯).
What do you notice? What does this tell us about ππ? Letβs look at the graph of ππ.
3. Consider the function ππ(π₯π₯) = 2π₯π₯3 β 5π₯π₯.
Find ππ(2) and ππ(β2). Find ππ(1) and ππ(β1).
Find ππ(βπ₯π₯)
What do you notice? What does this tell us about ππ? Letβs look at the graph of ππ.
Common Core Algebra II Lesson 14: Even and Odd Functions
59
4. Are all functions either even or odd? Consider the function ππ(π₯π₯) = π₯π₯2 β 4π₯π₯ + 1.
Find ππ(2) and ππ(β2). Find ππ(βπ₯π₯).
What can we conclude about ππ?
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
60
Homework 1. Half of the graph of ππ(π₯π₯) is shown below. Sketch the other half based on the function type.
2. Algebraically determine if ππ(π₯π₯) = 3π₯π₯4 β 7π₯π₯2 + 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
61
1. Your group will create a box from a piece of construction paper. Each group will record its boxβs 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 π₯π₯?
Write a function ππ(π₯π₯) that represents the volume of a box with a height of π₯π₯. 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
62
2. For a fundraiser, members of the math club decide to make and sell βPythagoras may have been Fermatβs first problem but not his last!β 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 π¦π¦-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
63
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 ππ(π₯π₯) = ππ(π₯π₯3 β 53π₯π₯2 β 236π₯π₯ + 9828). Use the point labeled on the graph to find the value of the leading coefficient, ππ.
g. Find ππ(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?
64
Opening Exercise
Compute 1355 Γ· 4 using long division.
1. Find each quotient by inspection.
π₯π₯+4π₯π₯+1
π₯π₯2+4π₯π₯2+6
2π₯π₯β7π₯π₯β3
2. Find the following quotient using both reverse tabular method and long division.
π₯π₯3 + 2π₯π₯2 + 8π₯π₯ + 1π₯π₯ + 5
Common Core Algebra II Lesson 16: What If There is a Remainder?
65
3. Find the following quotients using the method of your choice.
π₯π₯3βπ₯π₯2+3π₯π₯β1π₯π₯+3
π₯π₯2+4π₯π₯+10π₯π₯β8
4π₯π₯3+5π₯π₯β82π₯π₯β5
2π₯π₯3β4π₯π₯2β7π₯π₯β10π₯π₯β5
4π₯π₯2β5π₯π₯2β1
π₯π₯4β8π₯π₯2+12π₯π₯+2
Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 16: What If There is a Remainder?
66
Homework 1. Create equivalent expression in the form of ππ(π₯π₯) + ππ(π₯π₯)
ππ(π₯π₯) for each quotient.
π₯π₯3+7π₯π₯2+14π₯π₯+3π₯π₯+2
2π₯π₯2β13π₯π₯β10
2π₯π₯+3
π₯π₯β2π₯π₯+1
9π₯π₯3β12π₯π₯2+4
π₯π₯β2
2. Let ππ(π₯π₯) = π₯π₯3 + 2π₯π₯2 + 2π₯π₯ β 5. Divide ππ by π₯π₯ β 1. Then find ππ(1). Explain what you observe.
Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 17: The Remainder Theorem
67
Opening Exploration
Consider the polynomial function ππ(π₯π₯) = 3π₯π₯2 + 8π₯π₯ β 4.
a. Divide ππ by π₯π₯ β 2. b. Find ππ(2).
Consider the polynomial function ππ(π₯π₯) = π₯π₯3 β 3π₯π₯2 + 6π₯π₯ + 8.
a. Divide ππ by π₯π₯ + 1. b. Find ππ(β1).
Consider the polynomial β(π₯π₯) = π₯π₯3 + 2π₯π₯ β 3.
a. Divide β by π₯π₯ β 3. b. Find β(3).
Common Core Algebra II Lesson 17: The Remainder Theorem
68
The Remainder Theorem:
1. Determine if π₯π₯ β 4 is a factor of the function ππ(π₯π₯) = 2π₯π₯3 β 5π₯π₯2 β 11π₯π₯ β 4 using two different methods.
2. Determine if π₯π₯ + 2 is a factor of the function ππ(π₯π₯) = π₯π₯3 + π₯π₯2 β 27π₯π₯ β 15 using two different methods.
Common Core Algebra II Lesson 17: The Remainder Theorem
69
3. Consider the polynomial ππ(π₯π₯) + π₯π₯3 + πππ₯π₯2 + π₯π₯ + 6. Find the value of ππ so that π₯π₯ + 1 is a factor of ππ.
4. Find ππ(β1) if ππ(π₯π₯) = 2π₯π₯3 β 3π₯π₯2 β 17π₯π₯ β 12 and explain what your answer tells you about π₯π₯ + 1 as a factor of ππ.
5. Let π¦π¦ = ππ(π₯π₯) be the graph shown below. Find the remainder when ππ is divided by π₯π₯ β 1. Explain your reasoning.
Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 17: The Remainder Theorem
70
Homework 1. Let ππ(π₯π₯) = π₯π₯3 β 6π₯π₯2 β 7π₯π₯ + 9. Divide ππ by π₯π₯ β 3. Check your work by finding ππ(3).
2. Is π₯π₯ + 1 a factor of 2π₯π₯5 β 4π₯π₯4 + 9π₯π₯3 β π₯π₯ + 13? Explain your reasoning.
3. Which of the following binomials is a factor of π₯π₯3 + 3π₯π₯2 β 10π₯π₯ β 24.
(1) π₯π₯ β 1 (3) π₯π₯ β 3
(2) π₯π₯ β 2 (4) π₯π₯ β 4
4. The function ππ is defined by a polynomial. Some values of π₯π₯ and ππ(π₯π₯) are shown in the table below. What is the remainder when ππ is divide by π₯π₯ + 2? Explain your reasoning.
π₯π₯ ππ(π₯π₯) β3 β100 β2 β42 β1 β9 0 5 1 6 2 0 3 β7 4 β9
Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 18: Putting It All Together
71
Opening Exploration
Verify that π₯π₯ + 1 is a factor of ππ(π₯π₯) = 2π₯π₯3 + 3π₯π₯2 β 2π₯π₯ β 3. Explain your reasoning.
1. Knowing that π₯π₯ + 1 is factor of ππ, write ππ(π₯π₯) as the product of three linear factors.
State the zeros of ππ.
Sketch a graph of π¦π¦ = ππ(π₯π₯).
Common Core Algebra II Lesson 18: Putting It All Together
72
2. The polynomial function ππ(π₯π₯) = 2π₯π₯4 β 3π₯π₯3 β 15π₯π₯2 + 31π₯π₯ β 15 has a zero at π₯π₯ = 1 of multiplicity of 2.
Find all real zeros of ππ.
Describe the end behavior of ππ.
Sketch a graph of ππ showing the end behavior and zeros of ππ.
Common Core Algebra II Lesson 18: Putting It All Together
73
3. Given π§π§(π₯π₯) = 6π₯π₯3 + πππ₯π₯2 β 52π₯π₯ + 15, π§π§(2) = 35, and π§π§(β5) = 0, algebraically determine the zeros of π§π§(π₯π₯).
4. Consider the polynomial ππ(π₯π₯) = π₯π₯3 + πππ₯π₯2 β 10π₯π₯ + 24. Find the value of ππ so that π₯π₯ β 4 is a factor of ππ.
Express ππ as the product of three linear factors.
Name:_____________________________________________ Date:___________ Common Core Algebra II Lesson 18: Putting It All Together
74
Homework 1. Let ππ(π₯π₯) = 2π₯π₯4 + 11π₯π₯3 β 3π₯π₯2 β 44π₯π₯ β 20.
Explain why π₯π₯ + 5 is a factor of ππ.
Knowing π₯π₯ + 5 is a factor, factor ππ completely.
State the solutions to ππ(π₯π₯) = 0.
Describe the end behavior of ππ and sketch graph.
