Polynomial, rational, and radical relationships student classwork, homework, and templates

Published by Great Minds® Copyright © 2015 Great Minds No part of this work may be reproduced, sold, or commercialized, in whole or in part, without written permission from Great Minds Non commercial[.]

Published by Great Minds ®

Copyright © 2015 Great Minds No part of this work may be reproduced, sold, or commercialized, in whole or

in part, without written permission from Great Minds Non-commercial use is licensed pursuant to a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license; for more information, go to

http://greatminds.net/maps/math/copyright

Printed in the U.S.A

This book may be purchased from the publisher at eureka-math.org

10 9 8 7 6 5 4 3 2

Alg II-M1-SFA-1.3.2-06.2016

Algebra II Module 1 Student File_A

Student Workbook

This file contains

• Alg II-M1 Classwork

• Alg II-M1 Problem Sets

Lesson 1: Successive Differences in Polynomials

Assuming that the pattern would continue, he used it to find the value of 7 42 Explain how he used the pattern to find

7 42, and then use the pattern to find 8 42

How would you label each row of numbers in the table?

Discussion

Lesson 1: Successive Differences in Polynomials

Find the first, second, and third differences of the polynomial 𝑎𝑎𝑎𝑎2+ 𝑏𝑏𝑎𝑎 + 𝑐𝑐 by filling in the blanks in the following table

Lesson 1: Successive Differences in Polynomials

Find the equation of the form 𝑦𝑦 = 𝑎𝑎𝑎𝑎3+ 𝑏𝑏𝑎𝑎2+ 𝑐𝑐𝑎𝑎 + 𝑑𝑑 that all ordered pairs (𝑎𝑎, 𝑦𝑦) above satisfy Give evidence that your equation is correct

Relevant Vocabulary

N UMERICAL SYMBOL: A numerical symbol is a symbol that represents a specific number Examples: 1, 2, 3, 4, 𝜋𝜋, −3.2

V ARIABLE SYMBOL: A variable symbol is a symbol that is a placeholder for a number from a specified set of numbers The set of numbers is called the domain of the variable Examples: 𝑎𝑎, 𝑦𝑦, 𝑧𝑧

A LGEBRAIC EXPRESSION: An algebraic expression is either

1 a numerical symbol or a variable symbol or

2 the result of placing previously generated algebraic expressions into the two blanks of one of the four operators (( )+( ), ( )−( ), ( )×( ), ( )÷( )) or into the base blank of an exponentiation with an exponent that is a rational number

Following the definition above, ��(𝑎𝑎) × (𝑎𝑎)� × (𝑎𝑎)� + �(3) × (𝑎𝑎)� is an algebraic expression, but it is generally written more simply as 𝑎𝑎3+ 3𝑎𝑎

N UMERICAL EXPRESSION: A numerical expression is an algebraic expression that contains only numerical symbols (no

variable symbols) that evaluates to a single number Example: The numerical expression (3⋅2)

2

12 evaluates to 3

M ONOMIAL: A monomial is an algebraic expression generated using only the multiplication operator ( × ) The

expressions 𝑎𝑎3 and 3𝑎𝑎 are both monomials

B INOMIAL: A binomial is the sum of two monomials The expression 𝑎𝑎3+ 3𝑎𝑎 is a binomial

P OLYNOMIAL EXPRESSION: A polynomial expression is a monomial or sum of two or more monomials

S EQUENCE: A sequence can be thought of as an ordered list of elements The elements of the list are called the terms of

the sequence

A RITHMETIC SEQUENCE: A sequence is called arithmetic if there is a real number 𝑑𝑑 such that each term in the sequence is

the sum of the previous term and 𝑑𝑑

S.4

Problem Set

1 Create a table to find the second differences for the polynomial 36 − 16𝑡𝑡2 for integer values of 𝑡𝑡 from 0 to 5

2 Create a table to find the third differences for the polynomial 𝑠𝑠3− 𝑠𝑠2+ 𝑠𝑠 for integer values of 𝑠𝑠 from −3 to 3

3 Create a table of values for the polynomial 𝑎𝑎2, using 𝑛𝑛, 𝑛𝑛 + 1, 𝑛𝑛 + 2, 𝑛𝑛 + 3, 𝑛𝑛 + 4 as values of 𝑎𝑎 Show that the second differences are all equal to 2

4 Show that the set of ordered pairs (𝑎𝑎, 𝑦𝑦) in the table below satisfies a quadratic relationship (Hint: Find second differences.) Find the equation of the form 𝑦𝑦 = 𝑎𝑎𝑎𝑎2+ 𝑏𝑏𝑎𝑎 + 𝑐𝑐 that all of the ordered pairs satisfy

5 Show that the set of ordered pairs (𝑎𝑎, 𝑦𝑦) in the table below satisfies a cubic relationship (Hint: Find third

differences.) Find the equation of the form 𝑦𝑦 = 𝑎𝑎𝑎𝑎3+ 𝑏𝑏𝑎𝑎2+ 𝑐𝑐𝑎𝑎 + 𝑑𝑑 that all of the ordered pairs satisfy

a What type of relationship is indicated by the set of ordered pairs?

b Assuming that the relationship continues to hold, find the distance required to stop the car when the speed reaches 60 mph, when 𝑣𝑣 = 6

c Extension: Find an equation that describes the relationship between the speed of the car 𝑣𝑣 and its stopping distance 𝑑𝑑

7 Use the polynomial expressions 5𝑎𝑎2+ 𝑎𝑎 + 1 and 2𝑎𝑎 + 3 to answer the questions below

a Create a table of second differences for the polynomial 5𝑎𝑎2+ 𝑎𝑎 + 1 for the integer values of 𝑎𝑎 from 0 to 5

Lesson 2: The Multiplication of Polynomials

Lesson 2: The Multiplication of Polynomials

Classwork

Opening Exercise

Show that 28 × 27 = (20 + 8)(20 + 7) using an area model What do the numbers you placed inside the four

rectangular regions you drew represent?

Example 1

Use the tabular method to multiply (𝑎𝑎 + 8)(𝑎𝑎 + 7) and combine like terms

𝑎𝑎 + 8

𝑎𝑎 +

Exercises 1–2

1 Use the tabular method to multiply (𝑎𝑎2+ 3𝑎𝑎 + 1)(𝑎𝑎2− 5𝑎𝑎 + 2) and combine like terms

2 Use the tabular method to multiply (𝑎𝑎2+ 3𝑎𝑎 + 1)(𝑎𝑎2− 2) and combine like terms

Lesson 2: The Multiplication of Polynomials

Relevant Vocabulary

E QUIVALENT POLYNOMIAL EXPRESSIONS: Two polynomial expressions in one variable are equivalent if, whenever a number is

substituted into all instances of the variable symbol in both expressions, the numerical expressions created are equal

P OLYNOMIAL IDENTITY: A polynomial identity is a statement that two polynomial expressions are equivalent For example,

(𝑎𝑎 + 3)2= 𝑎𝑎2+ 6𝑎𝑎 + 9 for any real number 𝑎𝑎 is a polynomial identity

C OEFFICIENT OF A MONOMIAL: The coefficient of a monomial is the value of the numerical expression found by substituting

the number 1 into all the variable symbols in the monomial The coefficient of 3𝑎𝑎2 is 3, and the coefficient of the monomial (3𝑎𝑎𝑦𝑦𝑧𝑧) ⋅ 4 is 12

T ERMS OF A POLYNOMIAL : When a polynomial is expressed as a monomial or a sum of monomials, each monomial in the

sum is called a term of the polynomial

L IKE TERMS OF A POLYNOMIAL : Two terms of a polynomial that have the same variable symbols each raised to the same

power are called like terms

S TANDARD FORM OF A POLYNOMIAL IN ONE VARIABLE: A polynomial expression with one variable symbol, 𝑎𝑎, is in standard form

if it is expressed as

𝑎𝑎 𝑎𝑎 + 𝑎𝑎 1𝑎𝑎 1+ + 𝑎𝑎1𝑎𝑎 + 𝑎𝑎0, where 𝑛𝑛 is a non-negative integer, and 𝑎𝑎0, 𝑎𝑎1, 𝑎𝑎2…, 𝑎𝑎 are constant coefficients with 𝑎𝑎 0

A polynomial expression in 𝑎𝑎 that is in standard form is often just called a polynomial in 𝑎𝑎 or a polynomial

The degree of the polynomial in standard form is the highest degree of the terms in the polynomial, namely 𝑛𝑛 The term

𝑎𝑎 𝑎𝑎 is called the leading term and 𝑎𝑎 (thought of as a specific number) is called the leading coefficient The constant

term is the value of the numerical expression found by substituting 0 into all the variable symbols of the polynomial,

Lesson 2: The Multiplication of Polynomials

2 Use the tabular method to multiply and combine like terms

4 Polynomial expressions can be thought of as a generalization of place value

a Multiply 214 × 112 using the standard paper-and-pencil algorithm

b Multiply (2𝑎𝑎2+ 𝑎𝑎 + 4)(𝑎𝑎2+ 𝑎𝑎 + 2) using the tabular method and combine like terms

c Substitute 𝑎𝑎 = 10 into your answer from part (b)

d Is the answer to part (c) equal to the answer from part (a)? Compare the digits you computed in the algorithm

to the coefficients of the entries you computed in the table How do the place-value units of the digits compare to the powers of the variables in the entries?

5 Jeremy says (𝑎𝑎 − 9)(𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎3+ 𝑎𝑎2+ 𝑎𝑎 + 1) must equal 𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎3+ 𝑎𝑎2+ 𝑎𝑎 + 1 because when 𝑎𝑎 = 10, multiplying by 𝑎𝑎 − 9 is the same as multiplying by 1

a Multiply (𝑎𝑎 − 9)(𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎3+ 𝑎𝑎2+ 𝑎𝑎 + 1)

b Substitute 𝑎𝑎 = 10 into your answer

c Is the answer to part (b) the same as the value of 𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎 + 𝑎𝑎3+ 𝑎𝑎2+ 𝑎𝑎 + 1 when 𝑎𝑎 = 10?

d Was Jeremy right?

6 In the diagram, the side of the larger square is 𝑎𝑎 units, and the side of the

smaller square is 𝑦𝑦 units The area of the shaded region is (𝑎𝑎2− 𝑦𝑦2) square

units Show how the shaded area might be cut and rearranged to illustrate

that the area is (𝑎𝑎 − 𝑦𝑦)(𝑎𝑎 + 𝑦𝑦) square units

𝑎𝑎

𝑦𝑦

S.10

Lesson 3: The Division of Polynomials

Lesson 3: The Division of Polynomials

2 Describe the process you used to determine your answer to Exercise 1

3 Reverse the tabular method of multiplication to find the quotient: 2

5 Test your conjectures Use the reverse tabular method to find the quotient

3𝑎𝑎 − 2𝑎𝑎 + 6𝑎𝑎3− 4𝑎𝑎2− 24𝑎𝑎 + 16

𝑎𝑎2+ 4

1 ?

Lesson 3: The Division of Polynomials

Explain your prediction Then check your prediction using the reverse tabular method

Then check your prediction using the reverse tabular method

S.14

12 Consider the following quotients:

4𝑎𝑎2+ 8𝑎𝑎 + 3

48321

a How are these expressions related?

b Find each quotient

Lesson 4: Comparing Methods—Long Division, Again?

Lesson 4: Comparing Methods—Long Division, Again?

Example 1

If 𝑎𝑎 = 10, then the division 1573 ÷ 13 can be represented using polynomial division

3 7 5

3 3+ 2+ + + x x x x

Example 2

Use the long division algorithm for polynomials to evaluate

2𝑎𝑎3− 4𝑎𝑎2+ 2

Lesson 4: Comparing Methods—Long Division, Again?

Lesson 4: Comparing Methods—Long Division, Again?

a Determine the value of so that 3𝑎𝑎 − 7 is a factor of the polynomial

b What is the quotient when you divide the polynomial by 3𝑎𝑎 − 7?

Lesson Summary

The long division algorithm to divide polynomials is analogous to the long division algorithm for integers The long division algorithm to divide polynomials produces the same results as the reverse tabular method

S.20

8 In parts a–b and d–e, use long division to evaluate each quotient Then, answer the remaining questions

f Is 𝑎𝑎 + 3 a factor of 𝑎𝑎2+ 9? Explain your answer using the long division algorithm

g For which positive integers 𝑛𝑛 is 𝑎𝑎 + 3 a factor of 𝑎𝑎 + 3 ? Explain your reasoning

h If 𝑛𝑛 is a positive integer, is 𝑎𝑎 + 3 a factor of 𝑎𝑎 − 3 ? Explain your reasoning

Lesson 5: Putting It All Together

Lesson 5: Putting It All Together

Classwork

Exercises 1–15: Polynomial Pass

Perform the indicated operation to write each polynomial in standard form

15 (𝑎𝑎3+ 2𝑎𝑎2− 3𝑎𝑎 − 1) + (4 − 𝑎𝑎 − 𝑎𝑎3)

Lesson 5: Putting It All Together

Exercises 16–22

16 Review Exercises 1–15 and then select one exercise for each category and record the steps in the operation below

as an example Be sure to show all your work

S.24

For Exercises 17–20, rewrite each polynomial in standard form by applying the operations in the appropriate order

Lesson 5: Putting It All Together

21 What would be the first and last terms of the polynomial if it was rewritten in standard form? Answer these quickly without performing all of the indicated operations

What is the last term in the standard form of this polynomial? What does it mean in this situation?

11 Explain why these two quotients are different Compute each one What do they have in common? Why?

Lesson 6: Dividing by 𝑎𝑎 − 𝑎𝑎 and by 𝑎𝑎 + 𝑎𝑎

Lesson 6: Dividing by 𝒙𝒙 − 𝒂𝒂 and by 𝒙𝒙 + 𝒂𝒂

Lesson 6: Dividing by 𝑎𝑎 − 𝑎𝑎 and by 𝑎𝑎 + 𝑎𝑎

3 Predict without performing division whether or not the divisor will divide into the dividend without a remainder for the following problems If so, find the quotient Then check your answer

Lesson 6: Dividing by 𝑎𝑎 − 𝑎𝑎 and by 𝑎𝑎 + 𝑎𝑎

4

1 for 𝑛𝑛 = 2, 3, 4, and 8

b What patterns do you notice?

1 for any integer 𝑛𝑛 > 1

S.32

2 In the next exercises, you can use the same identities you applied in the previous problem Fill in the blanks in the problems below to help you get started Check your work by using the reverse tabular method or long division to make sure you are applying the identities correctly

𝑎𝑎 − 1 = (𝑎𝑎 − 1)(𝑎𝑎 1+ 𝑎𝑎 2+ 𝑎𝑎 3+ + 𝑎𝑎1+ 1), for integers 𝑛𝑛 > 1

Lesson 6: Dividing by 𝑎𝑎 − 𝑎𝑎 and by 𝑎𝑎 + 𝑎𝑎

3 Show how the patterns and relationships learned in this lesson could be applied to solve the following arithmetic problems by filling in the blanks

Lesson 7: Mental Math

Lesson 7: Mental Math

d 34 ⋅ 26

2 Find two additional factors of 2100− 1

3 Show that 83− 1 is divisible by 7

S.36

2 Give the general steps you take to determine 𝑎𝑎 and 𝑎𝑎 when asked to compute a product such as those in Problem 1

3 Why is 17 ⋅ 23 easier to compute than 17 ⋅ 22?

4 Rewrite the following differences of squares as a product of two integers

Lesson 7: Mental Math

9 Show that 999,973 is not prime without using a calculator or computer

10 Find a value of 𝑏𝑏 so that the expression 𝑏𝑏 − 1 is always divisible by 5 for any positive integer 𝑛𝑛 Explain why your value of 𝑏𝑏 works for any positive integer 𝑛𝑛

11 Find a value of 𝑏𝑏 so that the expression 𝑏𝑏 − 1 is always divisible by 7 for any positive integer 𝑛𝑛 Explain why your value of 𝑏𝑏 works for any positive integer 𝑛𝑛

12 Find a value of 𝑏𝑏 so that the expression 𝑏𝑏 − 1 is divisible by both 7 and 9 for any positive integer 𝑛𝑛 Explain why

your value of 𝑏𝑏 works for any positive integer 𝑛𝑛

S.38

Lesson 8: The Power of Algebra—Finding Primes

Classwork

Opening Exercise: When is 𝟐𝟐 − prime and when is it composite?

Complete the table to investigate which numbers of the form 2 − 1 are prime and which are composite

Lesson 8: The Power of Algebra—Finding Primes

Example 1: Proving a Conjecture

Start with an identity: 𝑎𝑎 − 1 = (𝑎𝑎 − 1)(𝑎𝑎 1+ 𝑎𝑎 2+ 𝑎𝑎1+ 1)

In this case, 𝑎𝑎 = 2, so the identity above becomes:

and it is not clear whether or not 2 − 1 is composite

Rewrite the expression: Let = 𝑎𝑎𝑏𝑏 be a positive odd composite number Then 𝑎𝑎 and 𝑏𝑏 must also be odd, or else

the product 𝑎𝑎𝑏𝑏 would be even The smallest such number is 9, so we have 𝑎𝑎 ≥ 3 and

Since 𝑎𝑎 ≥ 3, we have 2 ≥ 8; thus, 2 − 1 ≥ 7 Since the other factor is also larger than 1, 2 − 1 is

composite, and we have proven our conjecture

3 2 3 − 1 (Hint: 537 is the product of two prime numbers that are both less than 50.)

Exercise 4: How quickly can a computer factor a very large number?

4 How long would it take a computer to factor some squares of very large prime numbers?

The time in seconds required to factor an 𝑛𝑛-digit number of the form 2, where is a large prime, can roughly be approximated by (𝑛𝑛) = 3.4 × 10( 13)/2 Some values of this function are listed in the table below

Digits

Time needed to factor the number (sec)

Lesson 8: The Power of Algebra—Finding Primes

Problem Set

1 Factor 412− 1 in two different ways using the identity 𝑎𝑎 − 𝑎𝑎 = (𝑎𝑎 − 𝑎𝑎)(𝑎𝑎 + 𝑎𝑎𝑎𝑎 1+ 𝑎𝑎2𝑎𝑎 2+ + 𝑎𝑎 ) and the difference of squares identity

2 Factor 212+ 1 using the identity 𝑎𝑎 + 𝑎𝑎 = (𝑎𝑎 + 𝑎𝑎)(𝑎𝑎 − 𝑎𝑎𝑎𝑎 1+ 𝑎𝑎2𝑎𝑎 2− + 𝑎𝑎 ) for odd numbers 𝑛𝑛

3 Is 10,000,000,001 prime? Explain your reasoning

4 Explain why 2 − 1 is never prime if 𝑛𝑛 is a composite number

5 Fermat numbers are of the form 2 + 1 where 𝑛𝑛 is a positive integer

a Create a table of Fermat numbers for odd values of 𝑛𝑛 up to 9

b Explain why if 𝑛𝑛 is odd, the Fermat number 2 + 1 will always be divisible by 3

c Complete the table of values for even values of 𝑛𝑛 up to 12