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Printed in the U. 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-1.2016 A STORY OF FUNCTIONS Lesson 1 M1 ALGEBRA II Lesson 1: Successive Differences in Polynomials Classwork Opening Exercise John noticed patterns in the arrangement of numbers in the table below.8 2 2 2 Assuming that the pattern would continue, he used it to find the value of 7. Explain how he used the pattern to find 7. 42, and then use the pattern to find 8.
How would you label each row of numbers in the table? Discussion Let the sequence {𝑎𝑎0 , 𝑎𝑎1 , 𝑎𝑎 2, 𝑎𝑎 3, …} be generated by evaluating a polynomial expression at the values 0, 1, 2, 3, … The numbers found by evaluating 𝑎𝑎1 − 𝑎𝑎 0, 𝑎𝑎2 − 𝑎𝑎1 , 𝑎𝑎 3 − 𝑎𝑎 2, … form a new sequence, which we will call the first differences of the polynomial. The differences between successive terms of the first differences sequence are called the second differences, and so on. Lesson 1: Successive Differences in Polynomials S.org A STORY OF FUNCTIONS Lesson 1 M1 ALGEBRA II Example 1 What is the sequence of first differences for the linear polynomial given by 𝑎𝑎𝑎𝑎 + 𝑏𝑏, where 𝑎𝑎 and 𝑏𝑏 are constant coefficients? What is the sequence of second differences for 𝑎𝑎𝑎𝑎 + 𝑏𝑏? Example 2 Find the first, second, and third differences of the polynomial 𝑎𝑎𝑎𝑎 2 + 𝑏𝑏𝑎𝑎 + 𝑐𝑐 by filling in the blanks in the following table. 𝒙𝒙 𝒂𝒂𝒙𝒙𝟐𝟐 + 𝒃𝒃𝒙𝒙 + 𝒄𝒄 First Differences Second Differences Third Differences 0 𝑐𝑐 1 𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 2 4𝑎𝑎 + 2𝑏𝑏 + 𝑐𝑐 3 9𝑎𝑎 + 3𝑏𝑏 + 𝑐𝑐 4 16𝑎𝑎 + 4𝑏𝑏 + 𝑐𝑐 5 25𝑎𝑎 + 5𝑏𝑏 + 𝑐𝑐 Lesson 1: Successive Differences in Polynomials S.org A STORY OF FUNCTIONS Lesson 1 M1 ALGEBRA II Example 3 Find the second, third, and fourth differences of the polynomial 𝑎𝑎𝑎𝑎 3 + 𝑏𝑏𝑎𝑎 2 + 𝑐𝑐𝑎𝑎 + 𝑑𝑑 by filling in the blanks in the following table.
𝒙𝒙 𝒂𝒂𝒙𝒙 𝟑𝟑 + 𝒃𝒃𝒙𝒙𝟐𝟐 + 𝒄𝒄𝒙𝒙 + 𝒅𝒅 First Differences Second Differences Third Differences Fourth Differences 0 𝑑𝑑 𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 1 𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 + 𝑑𝑑 7𝑎𝑎 + 3𝑏𝑏 + 𝑐𝑐 2 8𝑎𝑎 + 4𝑏𝑏 + 2𝑐𝑐 + 𝑑𝑑 19𝑎𝑎 + 5𝑏𝑏 + 𝑐𝑐 3 27𝑎𝑎 + 9𝑏𝑏 + 3𝑐𝑐 + 𝑑𝑑 37𝑎𝑎 + 7𝑏𝑏 + 𝑐𝑐 4 64𝑎𝑎 + 16𝑏𝑏 + 4𝑐𝑐 + 𝑑𝑑 61𝑎𝑎 + 9𝑏𝑏 + 𝑐𝑐 5 125𝑎𝑎 + 25𝑏𝑏 + 5𝑐𝑐 + 𝑑𝑑 Example 4 What type of relationship does the set of ordered pairs (𝑎𝑎, 𝑦𝑦) satisfy? How do you know? Fill in the blanks in the table below to help you decide. (The first differences have already been computed for you.) 𝒙𝒙 𝒚𝒚 First Differences Second Differences Third Differences 0 2 −1 1 1 5 2 6 17 3 23 35 4 58 59 5 117 Lesson 1: Successive Differences in Polynomials S.org A STORY OF FUNCTIONS Lesson 1 M1 ALGEBRA II Find the equation of the form 𝑦𝑦 = 𝑎𝑎𝑎𝑎 3 + 𝑏𝑏𝑎𝑎 2 + 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.
VARIABLE 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. ALGEBRAIC 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 𝑎𝑎 3 + 3𝑎𝑎. N UMERICAL EXPRESSION: A numerical expression is an algebraic expression that contains only numerical symbols (no ( 3⋅2) 2 variable symbols) that evaluates to a single number. Example: The numerical expression evaluates to 3.
12 MONOMIAL: A monomial is an algebraic expression generated using only the multiplication operator (__×__). The expressions 𝑎𝑎 3 and 3𝑎𝑎 are both monomials. BINOMIAL: A binomial is the sum of two monomials. The expression 𝑎𝑎 3 + 3𝑎𝑎 is a binomial.
POLYNOMIAL EXPRESSION : A polynomial expression is a monomial or sum of two or more monomials. SEQUENCE: A sequence can be thought of as an ordered list of elements. The elements of the list are called the terms of the sequence. ARITHMETIC 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.
Lesson 1: Successive Differences in Polynomials S.org A STORY OF FUNCTIONS Lesson 1 M1 ALGEBRA II Problem Set 1. Create a table to find the second differences for the polynomial 36 − 16𝑡𝑡 2 for integer values of 𝑡𝑡 from 0 to 5. Create a table to find the third differences for the polynomial 𝑠𝑠 3 − 𝑠𝑠 2 + 𝑠𝑠 for integer values of 𝑠𝑠 from −3 to 3. Create a table of values for the polynomial 𝑎𝑎 2, using 𝑛𝑛, 𝑛𝑛 + 1, 𝑛𝑛 + 2, 𝑛𝑛 + 3, 𝑛𝑛 + 4 as values of 𝑎𝑎.
Show that the second differences are all equal to 2. 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 = 𝑎𝑎𝑎𝑎 2 + 𝑏𝑏𝑎𝑎 + 𝑐𝑐 that all of the ordered pairs satisfy. 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 = 𝑎𝑎𝑎𝑎 3 + 𝑏𝑏𝑎𝑎 2 + 𝑐𝑐𝑎𝑎 + 𝑑𝑑 that all of the ordered pairs satisfy. The distance 𝑑𝑑 ft. required to stop a car traveling at 10𝑣𝑣 mph under dry asphalt conditions is given by the following table. 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. Extension: Find an equation that describes the relationship between the speed of the car 𝑣𝑣 and its stopping distance 𝑑𝑑. Use the polynomial expressions 5𝑎𝑎 2 + 𝑎𝑎 + 1 and 2𝑎𝑎 + 3 to answer the questions below. Create a table of second differences for the polynomial 5𝑎𝑎 2 + 𝑎𝑎 + 1 for the integer values of 𝑎𝑎 from 0 to 5.
Justin claims that for 𝑛𝑛 ≥ 2, the 𝑛𝑛th differences of the sum of a degree 𝑛𝑛 polynomial and a linear polynomial are the same as the 𝑛𝑛th differences of just the degree 𝑛𝑛 polynomial. Find the second differences for the sum (5𝑎𝑎 2 + 𝑎𝑎 + 1) + (2𝑎𝑎 + 3) of a degree 2 and a degree 1 polynomial, and use the calculation to explain why Justin might be correct in general. Jason thinks he can generalize Justin’s claim to the product of two polynomials. He claims that for 𝑛𝑛 ≥ 2, the (𝑛𝑛 + 1)st differences of the product of a degree 𝑛𝑛 polynomial and a linear polynomial are the same as the 𝑛𝑛th differences of the degree 𝑛𝑛 polynomial.
Use what you know about second and third differences (from Examples 2 and 3) and the polynomial (5𝑎𝑎 2 + 𝑎𝑎 + 1)(2𝑎𝑎 + 3) to show that Jason’s generalization is incorrect. Lesson 1: Successive Differences in Polynomials S.org A STORY OF FUNCTIONS Lesson 2 M1 ALGEBRA II 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 2 8𝑎𝑎 𝑎𝑎 + 𝑎𝑎 2 7 7𝑎𝑎 56 15𝑎𝑎 56 Lesson 2: The Multiplication of Polynomials S.org A STORY OF FUNCTIONS Lesson 2 M1 ALGEBRA II Exercises 1–2 1.
Use the tabular method to multiply (𝑎𝑎 2 + 3𝑎𝑎 + 1)(𝑎𝑎 2 − 5𝑎𝑎 + 2) and combine like terms. Use the tabular method to multiply (𝑎𝑎 2 + 3𝑎𝑎 + 1)(𝑎𝑎 2 − 2) and combine like terms. Lesson 2: The Multiplication of Polynomials S.org A STORY OF FUNCTIONS Lesson 2 M1 ALGEBRA II Example 2 Multiply the polynomials (𝑎𝑎 − 1)(𝑎𝑎 + 𝑎𝑎 3 + 𝑎𝑎 2 + 𝑎𝑎 + 1) using a table. Generalize the pattern that emerges by writing down an identity for (𝑎𝑎 − 1)(𝑎𝑎 + 𝑎𝑎 1 + + 𝑎𝑎 2 + 𝑎𝑎 + 1) for 𝑛𝑛 a positive integer.
Multiply (𝑎𝑎 − 𝑦𝑦)(𝑎𝑎 3 + 𝑎𝑎 2 𝑦𝑦 + 𝑎𝑎𝑦𝑦 2 + 𝑦𝑦 3 ) using the distributive property and combine like terms. How is this calculation similar to Example 2? 4. Multiply (𝑎𝑎 2 − 𝑦𝑦 2 )(𝑎𝑎2 + 𝑦𝑦 2 ) using the distributive property and combine like terms. Generalize the pattern that emerges to write down an identity for (𝑎𝑎 − 𝑦𝑦 )(𝑎𝑎 + 𝑦𝑦 ) for positive integers 𝑛𝑛.
Lesson 2: The Multiplication of Polynomials S.org A STORY OF FUNCTIONS Lesson 2 M1 ALGEBRA II Relevant Vocabulary EQUIVALENT 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. POLYNOMIAL IDENTITY: A polynomial identity is a statement that two polynomial expressions are equivalent. For example, (𝑎𝑎 + 3) 2 = 𝑎𝑎 2 + 6𝑎𝑎 + 9 for any real number 𝑎𝑎 is a polynomial identity. COEFFICIENT 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 3𝑎𝑎 2 is 3, and the coefficient of the monomial (3𝑎𝑎𝑦𝑦𝑧𝑧) ⋅ 4 is 12. TERMS 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. STANDARD FORM OF A POLYNOMIAL IN ONE VARIABLE: A polynomial expression with one variable symbol, 𝑎𝑎, is in standard form if it is expressed as 1 𝑎𝑎 𝑎𝑎 + 𝑎𝑎 1 𝑎𝑎 + + 𝑎𝑎1 𝑎𝑎 + 𝑎𝑎 0, where 𝑛𝑛 is a non-negative integer, and 𝑎𝑎 0, 𝑎𝑎 1, 𝑎𝑎 2 …, 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, namely 𝑎𝑎 0.