Pre Calculus

NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S27 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties Classwork Exercise 1 Suzy draws the following picture to represent the sum 3 4 Ben looks at this picture from the opposite side of the table and says You drew 4 3 Explain why Ben might interpret the picture this way Exercise 2 Suzy adds more to her picture and says The picture now represents 3 4 2 How might Ben interpret this picture Explain your reasoning NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S28 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 3 Suzy then draws another picture of squares to represent the product 3 4 Ben moves to the end of the table and says From my new seat your picture looks like the product 4 3 What picture might Suzy have drawn Why would Ben see it differently from his viewpoint Exercise 4 Draw a picture to represent the quantity 3 4 5 that also could represent the quantity 4 5 3 when seen from a different viewpoint Four Properties of Arithmetic THE COMMUTATIVE PROPERTY OF ADDITION If 𝑎 and 𝑏 are real numbers then 𝑎 𝑏 𝑏 𝑎 THE ASSOCIATIVE PROPERTY OF ADDITION If 𝑎 𝑏 and 𝑐 are real numbers then 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 THE COMMUTATIVE PROPERTY OF MULTIPLICATION If 𝑎 and 𝑏 are real numbers then 𝑎 𝑏 𝑏 𝑎 THE ASSOCIATIVE PROPERTY OF MULTIPLICATION If 𝑎 𝑏 and 𝑐 are real numbers then 𝑎𝑏𝑐 𝑎𝑏𝑐 Exercise 5 Viewing the diagram below from two different perspectives illustrates that 3 4 2 equals 2 4 3 Is it true for all real numbers 𝑥 𝑦 and 𝑧 that 𝑥 𝑦 𝑧 should equal 𝑧 𝑦 𝑥 Note The direct application of the associative property of addition only gives 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S29 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 6 Draw a flow diagram and use it to prove that 𝑥𝑦𝑧 𝑧𝑦𝑥 for all real numbers 𝑥 𝑦 and 𝑧 Exercise 7 Use these abbreviations for the properties of real numbers and complete the flow diagram 𝐶 for the commutative property of addition 𝐶 for the commutative property of multiplication 𝐴 for the associative property of addition 𝐴 for the associative property of multiplication NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S30 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 8 Let 𝑎 𝑏 𝑐 and 𝑑 be real numbers Fill in the missing term of the following diagram to show that 𝑎 𝑏 𝑐 𝑑 is sure to equal 𝑎 𝑏 𝑐 𝑑 NUMERICAL SYMBOL A numerical symbol is a symbol that represents a specific number For example 0 1 2 3 2 3 3 124122 𝜋 𝑒 are numerical symbols used to represent specific points on the real number line VARIABLE SYMBOL A variable symbol is a symbol that is a placeholder for a number It is possible that a question may restrict the type of number that a placeholder might permit eg integers only or positive real numbers 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 Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the commutative associative and distributive properties and the properties of rational exponents to components of the first expression NUMERICAL EXPRESSION A numerical expression is an algebraic expression that contains only numerical symbols no variable symbols which evaluate to a single number The expression 3 0 is not a numerical expression EQUIVALENT NUMERICAL EXPRESSIONS Two numerical expressions are equivalent if they evaluate to the same number Note that 1 2 3 and 1 2 3 for example are equivalent numerical expressions they are both 6 but 𝑎 𝑏 𝑐 and 𝑎 𝑏 𝑐 are not equivalent expressions NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S31 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 The following portion of a flow diagram shows that the expression 𝑎𝑏 𝑐𝑑 is equivalent to the expression 𝑑𝑐 𝑏𝑎 Fill in each circle with the appropriate symbol Either 𝐶 for the commutative property of addition or 𝐶 for the commutative property of multiplication 2 Fill in the blanks of this proof showing that 𝑤 5𝑤 2 is equivalent to 𝑤2 7𝑤 10 Write either commutative property associative property or distributive property in each blank 𝑤 5𝑤 2 𝑤 5𝑤 𝑤 5 2 𝑤𝑤 5 𝑤 5 2 𝑤𝑤 5 2𝑤 5 𝑤2 𝑤 5 2𝑤 5 𝑤2 5𝑤 2𝑤 5 𝑤2 5𝑤 2𝑤 10 𝑤2 5𝑤 2𝑤 10 𝑤2 7𝑤 10 Lesson Summary The commutative and associative properties represent key beliefs about the arithmetic of real numbers These properties can be applied to algebraic expressions using variables that represent real numbers Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the commutative associative and distributive properties and the properties of rational exponents to components of the first expression NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S32 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Fill in each circle of the following flow diagram with one of the letters C for commutative property for either addition or multiplication A for associative property for either addition or multiplication or D for distributive property 4 What is a quick way to see that the value of the sum 53 18 47 82 is 200 5 a If 𝑎𝑏 37 and 1 37 what is the value of the product 𝑥 𝑏 𝑦 𝑎 b Give some indication as to how you used the commutative and associative properties of multiplication to evaluate 𝑥 𝑏 𝑦 𝑎 in part a c Did you use the associative and commutative properties of addition to answer Question 4 6 The following is a proof of the algebraic equivalency of 2𝑥3 and 8𝑥3 Fill in each of the blanks with either the statement commutative property or associative property 2𝑥3 2𝑥 2𝑥 2𝑥 2𝑥 2𝑥 2𝑥 22𝑥2𝑥𝑥 2 2𝑥 2𝑥 𝑥 2 22𝑥𝑥 𝑥 2 2 2𝑥 𝑥 𝑥 8𝑥3 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S33 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 7 Write a mathematical proof of the algebraic equivalency of 𝑎𝑏2 and 𝑎2𝑏 2 8 a Suppose we are to play the 4number game with the symbols 𝑎 𝑏 𝑐 and 𝑑 to represent numbers each used at most once combined by the operation of addition ONLY If we acknowledge that parentheses are unneeded show there are essentially only 15 expressions one can write b How many answers are there for the multiplication ONLY version of this game 9 Write a mathematical proof to show that 𝑥 𝑎𝑥 𝑏 is equivalent to 𝑥2 𝑎𝑥 𝑏𝑥 𝑎𝑏 10 Recall the following rules of exponents 𝑥𝑎 𝑥𝑏 𝑥𝑎𝑏 𝑥𝑎 𝑥𝑏 𝑥𝑎𝑏 𝑥𝑎𝑏 𝑥𝑎𝑏 𝑥𝑦𝑎 𝑥𝑎𝑦𝑎 𝑥 𝑦 𝑎 𝑥𝑎 𝑦𝑎 Here 𝑥 𝑦 𝑎 and 𝑏 are real numbers with 𝑥 and 𝑦 nonzero Replace each of the following expressions with an equivalent expression in which the variable of the expression appears only once with a positive number for its exponent For example 7 𝑏2 𝑏4 is equivalent to 7 𝑏6 a 16𝑥2 16𝑥5 b 2𝑥42𝑥3 c 9𝑧23𝑧13 d 25𝑤4 5𝑤3 5𝑤7 e 25𝑤4 5𝑤3 5𝑤7 Optional Challenge 11 Grizelda has invented a new operation that she calls the average operator For any two real numbers 𝑎 and 𝑏 she declares 𝑎 𝑏 to be the average of 𝑎 and 𝑏 𝑎 𝑏 𝑎 𝑏 2 a Does the average operator satisfy a commutative property That is does 𝑎 𝑏 𝑏 𝑎 for all real numbers 𝑎 and 𝑏 b Does the average operator distribute over addition That is does 𝑎𝑏 𝑐 𝑎𝑏 𝑎𝑐 for all real numbers 𝑎 𝑏 and 𝑐 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S34 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 8 Adding and Subtracting Polynomials Classwork Exercise 1 a How many quarters nickels and pennies are needed to make 113 b Fill in the blanks 8943 1000 100 10 1 103 102 10 1 c Fill in the blanks 8943 203 202 20 1 d Fill in the blanks 113 52 5 1 Exercise 2 Now lets be as general as possible by not identifying which base we are in Just call the base 𝑥 Consider the expression 1 𝑥3 2 𝑥2 7 𝑥 3 1 or equivalently 𝑥3 2𝑥2 7𝑥 3 a What is the value of this expression if 𝑥 10 b What is the value of this expression if 𝑥 20 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S35 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 3 a When writing numbers in base 10 we only allow coefficients of 0 through 9 Why is that b What is the value of 22𝑥 3 when 𝑥 5 How much money is 22 nickels and 3 pennies c What number is represented by 4𝑥2 17𝑥 2 if 𝑥 10 d What number is represented by 4𝑥2 17𝑥 2 if 𝑥 2 or if 𝑥 2 3 e What number is represented by 3𝑥2 2𝑥 1 2 when 𝑥 2 POLYNOMIAL EXPRESSION A polynomial expression is either 1 A numerical expression or a variable symbol or 2 The result of placing two previously generated polynomial expressions into the blanks of the addition operator or the multiplication operator NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S36 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 4 Find each sum or difference by combining the parts that are alike a 417 231 hundreds tens ones hundreds tens ones hundreds tens ones b 4𝑥2 𝑥 7 2𝑥2 3𝑥 1 c 3𝑥3 𝑥2 8 𝑥3 5𝑥2 4𝑥 7 d 3𝑥3 8𝑥 2𝑥3 12 e 5 𝑡 𝑡2 9𝑡 𝑡2 f 3𝑝 1 6𝑝 8 𝑝 2 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S37 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Celina says that each of the following expressions is actually a binomial in disguise i 5𝑎𝑏𝑐 2𝑎2 6𝑎𝑏𝑐 ii 5𝑥3 2𝑥2 10𝑥4 3𝑥5 3𝑥 2𝑥4 iii 𝑡 22 4𝑡 iv 5𝑎 1 10𝑎 1 100𝑎 1 v 2𝜋𝑟 𝜋𝑟2𝑟 2𝜋𝑟 𝜋𝑟2 2𝑟 For example she sees that the expression in i is algebraically equivalent to 11𝑎𝑏𝑐 2𝑎2 which is indeed a binomial She is happy to write this as 11𝑎𝑏𝑐 2𝑎2 if you prefer Is she right about the remaining four expressions 2 Janie writes a polynomial expression using only one variable 𝑥 with degree 3 Max writes a polynomial expression using only one variable 𝑥 with degree 7 a What can you determine about the degree of the sum of Janies and Maxs polynomials b What can you determine about the degree of the difference of Janies and Maxs polynomials 3 Suppose Janie writes a polynomial expression using only one variable 𝑥 with degree of 5 and Max writes a polynomial expression using only one variable 𝑥 with degree of 5 a What can you determine about the degree of the sum of Janies and Maxs polynomials b What can you determine about the degree of the difference of Janies and Maxs polynomials 4 Find each sum or difference by combining the parts that are alike a 2𝑝 4 5𝑝 1 𝑝 7 b 7𝑥4 9𝑥 2𝑥4 13 c 6 𝑡 𝑡4 9𝑡 𝑡4 d 5 𝑡2 6𝑡2 8 𝑡2 12 e 8𝑥3 5𝑥 3𝑥3 2 f 12𝑥 1 2𝑥 4 𝑥 15 g 13𝑥2 5𝑥 2𝑥2 1 h 9 𝑡 𝑡2 3 2 8𝑡 2𝑡2 i 4𝑚 6 12𝑚 3 𝑚 2 j 15𝑥4 10𝑥 12𝑥4 4𝑥 Lesson Summary A monomial is a polynomial expression generated using only the multiplication operator Thus it does not contain or operators Monomials are written with numerical factors multiplied together and variable or other symbols each occurring one time using exponents to condense multiple instances of the same variable A polynomial is the sum or difference of monomials The degree of a monomial is the sum of the exponents of the variable symbols that appear in the monomial The degree of a polynomial is the degree of the monomial term with the highest degree NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S119 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 24 Multiplying and Dividing Rational Expressions Classwork Example 1 Make a conjecture about the product 𝑥3 4𝑦 𝑦2 𝑥 What will it be Explain your conjecture and give evidence that it is correct Example 2 Find the following product 3𝑥 6 2𝑥 6 5𝑥 15 4𝑥 8 𝑎 𝑏 𝑐 𝑑 𝑎𝑐 𝑏𝑑 If 𝑎 𝑏 𝑐 and 𝑑 are rational expressions with 𝑏 0 𝑑 0 then NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S120 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercises 13 1 Summarize what you have learned so far with your neighbor 2 Find the following product and reduce to lowest terms 2𝑥6 𝑥2𝑥6 𝑥24 2𝑥 3 Find the following product and reduce to lowest terms 4𝑛12 3𝑚6 2 𝑛22𝑛3 𝑚24𝑚4 𝑎 𝑏 𝑐 𝑑 𝑎 𝑏 𝑑 𝑐 𝑎𝑑 𝑏𝑐 If 𝑎 𝑏 𝑐 and 𝑑 are rational expressions with 𝑏 0 𝑐 0 and 𝑑 0 then NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S121 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Example 3 Find the quotient and reduce to lowest terms 𝑥2 4 3𝑥 𝑥 2 2𝑥 Exercises 45 4 Find the quotient and reduce to lowest terms 𝑥2 5𝑥 6 𝑥 4 𝑥2 9 𝑥2 5𝑥 4 5 Simplify the rational expression 𝑥 2 𝑥2 2𝑥 3 𝑥2 𝑥 6 𝑥2 6𝑥 5 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S122 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Perform the following operations a Multiply 1 3 𝑥 2 by 9 b Divide 1 4 𝑥 8 by 1 12 c Multiply 1 4 1 3 𝑥 2 by 12 d Divide 1 3 2 5 𝑥 1 5 by 1 15 e Multiply 2 3 2𝑥 2 3 by 9 4 f Multiply 0034 𝑥 by 100 2 Write each rational expression as an equivalent rational expression in lowest terms a 𝑎3𝑏2 𝑐2𝑑2 𝑐 𝑎𝑏 𝑎 𝑐2𝑑3 b 𝑎26𝑎9 𝑎29 3𝑎9 𝑎3 c 6𝑥 4𝑥16 4𝑥 𝑥216 d 3𝑥26𝑥 3𝑥1 𝑥3𝑥2 𝑥24𝑥4 e 2𝑥210𝑥12 𝑥24 2𝑥 3𝑥 f 𝑎2𝑏 𝑎2𝑏 4𝑏2 𝑎2 g 𝑑𝑐 𝑐2𝑑2 𝑐2𝑑2 𝑑2𝑑𝑐 h 12𝑎27𝑎𝑏𝑏2 9𝑎2𝑏2 16𝑎2𝑏2 3𝑎𝑏𝑏2 i 𝑥3 𝑥24 1 𝑥2𝑥6 𝑥2 j 𝑥2 𝑥21 3 𝑥24𝑥4 𝑥22𝑥3 k 6𝑥211𝑥10 6𝑥25𝑥6 64𝑥 2520𝑥4𝑥2 l 3𝑥33𝑎2𝑥 𝑥22𝑎𝑥𝑎2 𝑎𝑥 𝑎3𝑥𝑎2𝑥2 Lesson Summary In this lesson we extended multiplication and division of rational numbers to multiplication and division of rational expressions To multiply two rational expressions multiply the numerators together and multiply the denominators together and then reduce to lowest terms To divide one rational expression by another multiply the first by the multiplicative inverse of the second and reduce to lowest terms To simplify a complex fraction apply the process for dividing one rational expression by another NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S123 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Write each rational expression as an equivalent rational expression in lowest terms a 4𝑎 6𝑏2 20𝑎3 12𝑏 b 𝑥2 𝑥21 𝑥24 𝑥6 c 𝑥22𝑥3 𝑥23𝑥4 𝑥2𝑥6 𝑥4 4 Suppose that 𝑥 𝑡23𝑡4 3𝑡23 and 𝑦 𝑡22𝑡8 2𝑡22𝑡4 for 𝑡 1 𝑡 1 𝑡 2 and 𝑡 4 Show that the value of 𝑥2𝑦2 does not depend on the value of 𝑡 5 Determine which of the following numbers is larger without using a calculator 1516 1615 or 2024 2420 Hint We can compare two positive quantities 𝑎 and 𝑏 by computing the quotient 𝑎 𝑏 If 𝑎 𝑏 1 then 𝑎 𝑏 Likewise if 0 𝑎 𝑏 1 then 𝑎 𝑏 Extension 6 One of two numbers can be represented by the rational expression 𝑥2 𝑥 where 𝑥 0 and 𝑥 2 a Find a representation of the second number if the product of the two numbers is 1 b Find a representation of the second number if the product of the two numbers is 0 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S34 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 8 Adding and Subtracting Polynomials Classwork Exercise 1 a How many quarters nickels and pennies are needed to make 113 b Fill in the blanks 8943 1000 100 10 1 103 102 10 1 c Fill in the blanks 8943 203 202 20 1 d Fill in the blanks 113 52 5 1 Exercise 2 Now lets be as general as possible by not identifying which base we are in Just call the base 𝑥 Consider the expression 1 𝑥3 2 𝑥2 7 𝑥 3 1 or equivalently 𝑥3 2𝑥2 7𝑥 3 a What is the value of this expression if 𝑥 10 b What is the value of this expression if 𝑥 20 4 quarters 2 nickels 3 pennies 8 9 Y 3 G e 4 3 2 7 3 4 2 3 10 2 102 7 10 3 1000 2 100 70 3 1273 20 2 20 2 7 20 3 8000 2 400 140 3 8000 800 140 3 8943 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S35 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 3 a When writing numbers in base 10 we only allow coefficients of 0 through 9 Why is that b What is the value of 22𝑥 3 when 𝑥 5 How much money is 22 nickels and 3 pennies c What number is represented by 4𝑥2 17𝑥 2 if 𝑥 10 d What number is represented by 4𝑥2 17𝑥 2 if 𝑥 2 or if 𝑥 2 3 e What number is represented by 3𝑥2 2𝑥 1 2 when 𝑥 2 POLYNOMIAL EXPRESSION A polynomial expression is either 1 A numerical expression or a variable symbol or 2 The result of placing two previously generated polynomial expressions into the blanks of the addition operator or the multiplication operator Base 10 uses digits O through a for simplicity in positional value and arithmetic operations This system aligns with human counting practices and ensures global compatibility Using just these ten digits avoids complexity and keeps calculations straight forward 22 5 3 10 3 113 1 13 4 102 17 10 2 4 100 170 2 400 170 2 572 4 21 2 17 21 2 4 8 2 178 2 y y 1 34 2 4 16 34 2 4 16 1 312 2 2 1 2 3 2 2 1 2 6 6 2 1 12 4 1 7 2 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S36 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 4 Find each sum or difference by combining the parts that are alike a 417 231 hundreds tens ones hundreds tens ones hundreds tens ones b 4𝑥2 𝑥 7 2𝑥2 3𝑥 1 c 3𝑥3 𝑥2 8 𝑥3 5𝑥2 4𝑥 7 d 3𝑥3 8𝑥 2𝑥3 12 e 5 𝑡 𝑡2 9𝑡 𝑡2 f 3𝑝 1 6𝑝 8 𝑝 2 4 I 7 2 3 1 648 6 Y 8 Yx x 7 2x 3x 1 6x2 4x 8 3x x2 8 x3 5x 4x 7 2x 6x 4x 15 3x3 24x 2x 24 x3 24x 24 5 t ta 9t 2 8t 5 3p 1 6p 48 p 2 48p 49 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S37 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Celina says that each of the following expressions is actually a binomial in disguise i 5𝑎𝑏𝑐 2𝑎2 6𝑎𝑏𝑐 ii 5𝑥3 2𝑥2 10𝑥4 3𝑥5 3𝑥 2𝑥4 iii 𝑡 22 4𝑡 iv 5𝑎 1 10𝑎 1 100𝑎 1 v 2𝜋𝑟 𝜋𝑟2𝑟 2𝜋𝑟 𝜋𝑟2 2𝑟 For example she sees that the expression in i is algebraically equivalent to 11𝑎𝑏𝑐 2𝑎2 which is indeed a binomial She is happy to write this as 11𝑎𝑏𝑐 2𝑎2 if you prefer Is she right about the remaining four expressions 2 Janie writes a polynomial expression using only one variable 𝑥 with degree 3 Max writes a polynomial expression using only one variable 𝑥 with degree 7 a What can you determine about the degree of the sum of Janies and Maxs polynomials b What can you determine about the degree of the difference of Janies and Maxs polynomials 3 Suppose Janie writes a polynomial expression using only one variable 𝑥 with degree of 5 and Max writes a polynomial expression using only one variable 𝑥 with degree of 5 a What can you determine about the degree of the sum of Janies and Maxs polynomials b What can you determine about the degree of the difference of Janies and Maxs polynomials 4 Find each sum or difference by combining the parts that are alike a 2𝑝 4 5𝑝 1 𝑝 7 b 7𝑥4 9𝑥 2𝑥4 13 c 6 𝑡 𝑡4 9𝑡 𝑡4 d 5 𝑡2 6𝑡2 8 𝑡2 12 e 8𝑥3 5𝑥 3𝑥3 2 f 12𝑥 1 2𝑥 4 𝑥 15 g 13𝑥2 5𝑥 2𝑥2 1 h 9 𝑡 𝑡2 3 2 8𝑡 2𝑡2 i 4𝑚 6 12𝑚 3 𝑚 2 j 15𝑥4 10𝑥 12𝑥4 4𝑥 Lesson Summary A monomial is a polynomial expression generated using only the multiplication operator Thus it does not contain or operators Monomials are written with numerical factors multiplied together and variable or other symbols each occurring one time using exponents to condense multiple instances of the same variable A polynomial is the sum or difference of monomials The degree of a monomial is the sum of the exponents of the variable symbols that appear in the monomial The degree of a polynomial is the degree of the monomial term with the highest degree COx10x 3x5 6x 07x 10x 7x 10x ta 2t 2t y 47 2 y 72 1 4 52 5 10a 10 1002 100 95a 95 952 95 952 1 2 rerHir Zirr2ar 2 Ir2 ir Yes 2 al Px Janies 3 Px Qx x Maxs 7 max 3 71 7 The degree is 7 b PxQlx max 13 7 7 0 The degree is 7 3 al P 5 PlecQu Qx max 5 51 5 The degree is 5 or less because the terms can be canceled out b Px 5 Px Qx Qx max 5 5 5 The degree is 5 or less because the terms can be canceled out a 2p 4 5p 5 p 7 bp 8 b Fx 9x 2x 26 5x 9x 26 c 6 t t 9t t 8t 6 d 5ta 6t2 48 t2 12 4t2 43 el 8x 5x 3x 6 05x 5x 6 f2x 1 2x 8 x 15 13x 8 g 13x4 5x 2x 2 x 5x 2 h 9 t t2 12t 372 yt 13 9 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S27 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties Classwork Exercise 1 Suzy draws the following picture to represent the sum 3 4 Ben looks at this picture from the opposite side of the table and says You drew 4 3 Explain why Ben might interpret the picture this way Exercise 2 Suzy adds more to her picture and says The picture now represents 3 4 2 How might Ben interpret this picture Explain your reasoning Ben is looking from the opposite side like that So he sees it on the opposite order Ben Suzy too 2 4 3 Ben is looking from the opposite side like that So he sees it on ⑭ the opposite order Suzy Ben too NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S28 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 3 Suzy then draws another picture of squares to represent the product 3 4 Ben moves to the end of the table and says From my new seat your picture looks like the product 4 3 What picture might Suzy have drawn Why would Ben see it differently from his viewpoint Exercise 4 Draw a picture to represent the quantity 3 4 5 that also could represent the quantity 4 5 3 when seen from a different viewpoint Four Properties of Arithmetic THE COMMUTATIVE PROPERTY OF ADDITION If 𝑎 and 𝑏 are real numbers then 𝑎 𝑏 𝑏 𝑎 THE ASSOCIATIVE PROPERTY OF ADDITION If 𝑎 𝑏 and 𝑐 are real numbers then 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 THE COMMUTATIVE PROPERTY OF MULTIPLICATION If 𝑎 and 𝑏 are real numbers then 𝑎 𝑏 𝑏 𝑎 THE ASSOCIATIVE PROPERTY OF MULTIPLICATION If 𝑎 𝑏 and 𝑐 are real numbers then 𝑎𝑏𝑐 𝑎𝑏𝑐 Exercise 5 Viewing the diagram below from two different perspectives illustrates that 3 4 2 equals 2 4 3 Is it true for all real numbers 𝑥 𝑦 and 𝑧 that 𝑥 𝑦 𝑧 should equal 𝑧 𝑦 𝑥 Note The direct application of the associative property of addition only gives 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 Suzy Ben Yes NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S29 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 6 Draw a flow diagram and use it to prove that 𝑥𝑦𝑧 𝑧𝑦𝑥 for all real numbers 𝑥 𝑦 and 𝑧 Exercise 7 Use these abbreviations for the properties of real numbers and complete the flow diagram 𝐶 for the commutative property of addition 𝐶 for the commutative property of multiplication 𝐴 for the associative property of addition 𝐴 for the associative property of multiplication O xy z A xyz yx t zyk 2x Cx C C C NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S30 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 8 Let 𝑎 𝑏 𝑐 and 𝑑 be real numbers Fill in the missing term of the following diagram to show that 𝑎 𝑏 𝑐 𝑑 is sure to equal 𝑎 𝑏 𝑐 𝑑 NUMERICAL SYMBOL A numerical symbol is a symbol that represents a specific number For example 0 1 2 3 2 3 3 124122 𝜋 𝑒 are numerical symbols used to represent specific points on the real number line VARIABLE SYMBOL A variable symbol is a symbol that is a placeholder for a number It is possible that a question may restrict the type of number that a placeholder might permit eg integers only or positive real numbers 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 Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the commutative associative and distributive properties and the properties of rational exponents to components of the first expression NUMERICAL EXPRESSION A numerical expression is an algebraic expression that contains only numerical symbols no variable symbols which evaluate to a single number The expression 3 0 is not a numerical expression EQUIVALENT NUMERICAL EXPRESSIONS Two numerical expressions are equivalent if they evaluate to the same number Note that 1 2 3 and 1 2 3 for example are equivalent numerical expressions they are both 6 but 𝑎 𝑏 𝑐 and 𝑎 𝑏 𝑐 are not equivalent expressions allb c d NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S31 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 The following portion of a flow diagram shows that the expression 𝑎𝑏 𝑐𝑑 is equivalent to the expression 𝑑𝑐 𝑏𝑎 Fill in each circle with the appropriate symbol Either 𝐶 for the commutative property of addition or 𝐶 for the commutative property of multiplication 2 Fill in the blanks of this proof showing that 𝑤 5𝑤 2 is equivalent to 𝑤2 7𝑤 10 Write either commutative property associative property or distributive property in each blank 𝑤 5𝑤 2 𝑤 5𝑤 𝑤 5 2 𝑤𝑤 5 𝑤 5 2 𝑤𝑤 5 2𝑤 5 𝑤2 𝑤 5 2𝑤 5 𝑤2 5𝑤 2𝑤 5 𝑤2 5𝑤 2𝑤 10 𝑤2 5𝑤 2𝑤 10 𝑤2 7𝑤 10 Lesson Summary The commutative and associative properties represent key beliefs about the arithmetic of real numbers These properties can be applied to algebraic expressions using variables that represent real numbers Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the commutative associative and distributive properties and the properties of rational exponents to components of the first expression C Cx Ex Commutative multiplication Commutative multiplication Distributive multiplication Associative multiplication Distributive multiplication Commutative addition Associative addition NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S32 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Fill in each circle of the following flow diagram with one of the letters C for commutative property for either addition or multiplication A for associative property for either addition or multiplication or D for distributive property 4 What is a quick way to see that the value of the sum 53 18 47 82 is 200 5 a If 𝑎𝑏 37 and 1 37 what is the value of the product 𝑥 𝑏 𝑦 𝑎 b Give some indication as to how you used the commutative and associative properties of multiplication to evaluate 𝑥 𝑏 𝑦 𝑎 in part a c Did you use the associative and commutative properties of addition to answer Question 4 6 The following is a proof of the algebraic equivalency of 2𝑥3 and 8𝑥3 Fill in each of the blanks with either the statement commutative property or associative property 2𝑥3 2𝑥 2𝑥 2𝑥 2𝑥 2𝑥 2𝑥 22𝑥2𝑥𝑥 2 2𝑥 2𝑥 𝑥 2 22𝑥𝑥 𝑥 2 2 2𝑥 𝑥 𝑥 8𝑥3 C C D C C C A D D C C 53 47 100 18782 100 100 100 200 37 1 x b y a y b 2 137 5 Yes Associative commutative Associative commutative Associative NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S33 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 7 Write a mathematical proof of the algebraic equivalency of 𝑎𝑏2 and 𝑎2𝑏 2 8 a Suppose we are to play the 4number game with the symbols 𝑎 𝑏 𝑐 and 𝑑 to represent numbers each used at most once combined by the operation of addition ONLY If we acknowledge that parentheses are unneeded show there are essentially only 15 expressions one can write b How many answers are there for the multiplication ONLY version of this game 9 Write a mathematical proof to show that 𝑥 𝑎𝑥 𝑏 is equivalent to 𝑥2 𝑎𝑥 𝑏𝑥 𝑎𝑏 10 Recall the following rules of exponents 𝑥𝑎 𝑥𝑏 𝑥𝑎𝑏 𝑥𝑎 𝑥𝑏 𝑥𝑎𝑏 𝑥𝑎𝑏 𝑥𝑎𝑏 𝑥𝑦𝑎 𝑥𝑎𝑦𝑎 𝑥 𝑦 𝑎 𝑥𝑎 𝑦𝑎 Here 𝑥 𝑦 𝑎 and 𝑏 are real numbers with 𝑥 and 𝑦 nonzero Replace each of the following expressions with an equivalent expression in which the variable of the expression appears only once with a positive number for its exponent For example 7 𝑏2 𝑏4 is equivalent to 7 𝑏6 a 16𝑥2 16𝑥5 b 2𝑥42𝑥3 c 9𝑧23𝑧13 d 25𝑤4 5𝑤3 5𝑤7 e 25𝑤4 5𝑤3 5𝑤7 Optional Challenge 11 Grizelda has invented a new operation that she calls the average operator For any two real numbers 𝑎 and 𝑏 she declares 𝑎 𝑏 to be the average of 𝑎 and 𝑏 𝑎 𝑏 𝑎 𝑏 2 a Does the average operator satisfy a commutative property That is does 𝑎 𝑏 𝑏 𝑎 for all real numbers 𝑎 and 𝑏 b Does the average operator distribute over addition That is does 𝑎𝑏 𝑐 𝑎𝑏 𝑎𝑐 for all real numbers 𝑎 𝑏 and 𝑐 labl lablabl 2 b a b a 2 b b 22b2 x 4 bx 2x ab 8 ala b 2 d Sterm Yoptions a b a c a dib c b d c d 2 terms 6 options 2 b ca b d b c d a c d Sterms 4 options a b c d Y terms I option 4 6 4 1 15 b 2 b c d Y options 2 b a cia dib c b d cd6 options a b c 2 bid a cd b cdo Y options a D c d o I option 4 6 y 1 25 10 al 16x2 x 3 b 2x4 3 2x1 16x5 I I 2 192 37437 3z1z 3 al 252 5rS 25 5wS S 5w el 25w 25 y Y 25 b Jus wso 25w 5w NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S34 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 8 Adding and Subtracting Polynomials Classwork Exercise 1 a How many quarters nickels and pennies are needed to make 113 b Fill in the blanks 8943 1000 100 10 1 103 102 10 1 c Fill in the blanks 8943 203 202 20 1 d Fill in the blanks 113 52 5 1 Exercise 2 Now lets be as general as possible by not identifying which base we are in Just call the base 𝑥 Consider the expression 1 𝑥3 2 𝑥2 7 𝑥 3 1 or equivalently 𝑥3 2𝑥2 7𝑥 3 a What is the value of this expression if 𝑥 10 b What is the value of this expression if 𝑥 20 4 quarters 2 nickels 3 pennies 8 9 Y 3 G e 4 3 2 7 3 4 2 3 10 2 102 7 10 3 1000 2 100 70 3 1273 20 2 20 2 7 20 3 8000 2 400 140 3 8000 800 140 3 8943 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S35 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 3 a When writing numbers in base 10 we only allow coefficients of 0 through 9 Why is that b What is the value of 22𝑥 3 when 𝑥 5 How much money is 22 nickels and 3 pennies c What number is represented by 4𝑥2 17𝑥 2 if 𝑥 10 d What number is represented by 4𝑥2 17𝑥 2 if 𝑥 2 or if 𝑥 2 3 e What number is represented by 3𝑥2 2𝑥 1 2 when 𝑥 2 POLYNOMIAL EXPRESSION A polynomial expression is either 1 A numerical expression or a variable symbol or 2 The result of placing two previously generated polynomial expressions into the blanks of the addition operator or the multiplication operator Base 10 uses digits O through a for simplicity in positional value and arithmetic operations This system aligns with human counting practices and ensures global compatibility Using just these ten digits avoids complexity and keeps calculations straight forward 22 5 3 10 3 113 1 13 4 102 17 10 2 4 100 170 2 400 170 2 572 4 21 2 17 21 2 4 8 2 178 2 y y 1 34 2 4 16 34 2 4 16 1 312 2 2 1 2 3 2 2 1 2 6 6 2 1 12 4 1 7 2 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S36 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 4 Find each sum or difference by combining the parts that are alike a 417 231 hundreds tens ones hundreds tens ones hundreds tens ones b 4𝑥2 𝑥 7 2𝑥2 3𝑥 1 c 3𝑥3 𝑥2 8 𝑥3 5𝑥2 4𝑥 7 d 3𝑥3 8𝑥 2𝑥3 12 e 5 𝑡 𝑡2 9𝑡 𝑡2 f 3𝑝 1 6𝑝 8 𝑝 2 4 I 7 2 3 1 648 6 Y 8 Yx x 7 2x 3x 1 6x2 4x 8 3x x2 8 x3 5x 4x 7 2x 6x 4x 15 3x3 24x 2x 24 x3 24x 24 5 t ta 9t 2 8t 5 3p 1 6p 48 p 2 48p 49 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 8 ALGEBRA I Lesson 8 Adding and Subtracting Polynomials S37 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Celina says that each of the following expressions is actually a binomial in disguise i 5𝑎𝑏𝑐 2𝑎2 6𝑎𝑏𝑐 ii 5𝑥3 2𝑥2 10𝑥4 3𝑥5 3𝑥 2𝑥4 iii 𝑡 22 4𝑡 iv 5𝑎 1 10𝑎 1 100𝑎 1 v 2𝜋𝑟 𝜋𝑟2𝑟 2𝜋𝑟 𝜋𝑟2 2𝑟 For example she sees that the expression in i is algebraically equivalent to 11𝑎𝑏𝑐 2𝑎2 which is indeed a binomial She is happy to write this as 11𝑎𝑏𝑐 2𝑎2 if you prefer Is she right about the remaining four expressions 2 Janie writes a polynomial expression using only one variable 𝑥 with degree 3 Max writes a polynomial expression using only one variable 𝑥 with degree 7 a What can you determine about the degree of the sum of Janies and Maxs polynomials b What can you determine about the degree of the difference of Janies and Maxs polynomials 3 Suppose Janie writes a polynomial expression using only one variable 𝑥 with degree of 5 and Max writes a polynomial expression using only one variable 𝑥 with degree of 5 a What can you determine about the degree of the sum of Janies and Maxs polynomials b What can you determine about the degree of the difference of Janies and Maxs polynomials 4 Find each sum or difference by combining the parts that are alike a 2𝑝 4 5𝑝 1 𝑝 7 b 7𝑥4 9𝑥 2𝑥4 13 c 6 𝑡 𝑡4 9𝑡 𝑡4 d 5 𝑡2 6𝑡2 8 𝑡2 12 e 8𝑥3 5𝑥 3𝑥3 2 f 12𝑥 1 2𝑥 4 𝑥 15 g 13𝑥2 5𝑥 2𝑥2 1 h 9 𝑡 𝑡2 3 2 8𝑡 2𝑡2 i 4𝑚 6 12𝑚 3 𝑚 2 j 15𝑥4 10𝑥 12𝑥4 4𝑥 Lesson Summary A monomial is a polynomial expression generated using only the multiplication operator Thus it does not contain or operators Monomials are written with numerical factors multiplied together and variable or other symbols each occurring one time using exponents to condense multiple instances of the same variable A polynomial is the sum or difference of monomials The degree of a monomial is the sum of the exponents of the variable symbols that appear in the monomial The degree of a polynomial is the degree of the monomial term with the highest degree COx10x 3x5 6x 07x 10x 7x 10x ta 2t 2t y 47 2 y 72 1 4 52 5 10a 10 1002 100 95a 95 952 95 952 1 2 rerHir Zirr2ar 2 Ir2 ir Yes 2 al Px Janies 3 Px Qx x Maxs 7 max 3 71 7 The degree is 7 b PxQlx max 13 7 7 0 The degree is 7 3 al P 5 PlecQu Qx max 5 51 5 The degree is 5 or less because the terms can be canceled out b Px 5 Px Qx Qx max 5 5 5 The degree is 5 or less because the terms can be canceled out a 2p 4 5p 5 p 7 bp 8 b Fx 9x 2x 26 5x 9x 26 c 6 t t 9t t 8t 6 d 5ta 6t2 48 t2 12 4t2 43 el 8x 5x 3x 6 05x 5x 6 f2x 1 2x 8 x 15 13x 8 g 13x4 5x 2x 2 x 5x 2 h 9 t t2 12t 372 yt 13 9 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S119 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 24 Multiplying and Dividing Rational Expressions Classwork Example 1 Make a conjecture about the product 𝑥3 4𝑦 𝑦2 𝑥 What will it be Explain your conjecture and give evidence that it is correct Example 2 Find the following product 3𝑥 6 2𝑥 6 5𝑥 15 4𝑥 8 𝑎 𝑏 𝑐 𝑑 𝑎𝑐 𝑏𝑑 If 𝑎 𝑏 𝑐 and 𝑑 are rational expressions with 𝑏 0 𝑑 0 then 3 2 the process of multiplying and r y sy ray simplifying shows that this expression 4y x 4 is correct Each step involved using algebric rules such as multiplying fractions canceling common terms and simplifying the result which confirms that the Final expression is valid 15x2 45x 30x 90 8x 4 16x 24x 48 15x2 15x 90 8x2 40x 48 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S120 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercises 13 1 Summarize what you have learned so far with your neighbor 2 Find the following product and reduce to lowest terms 2𝑥6 𝑥2𝑥6 𝑥24 2𝑥 3 Find the following product and reduce to lowest terms 4𝑛12 3𝑚6 2 𝑛22𝑛3 𝑚24𝑚4 𝑎 𝑏 𝑐 𝑑 𝑎 𝑏 𝑑 𝑐 𝑎𝑑 𝑏𝑐 If 𝑎 𝑏 𝑐 and 𝑑 are rational expressions with 𝑏 0 𝑐 0 and 𝑑 0 then 2x 8x 6x2 24 2x 2x 12x 2x 6x Ex24 2x 2x2 12x 11 Eme 2 3m3m b 9m 18m 18m 36 X 4n 12 Yn 12 Yn 12 1n2 48n 48u 144 9m2 36m 36 n 2 2n 3 Emen 18mn27m 36mn72mn108m 36n72n108 I 16n2 96n 144 ma ym 4 1 nim 64ndm 64na 964m2 384nm 38yn 144m2 576m 576 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S121 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Example 3 Find the quotient and reduce to lowest terms 𝑥2 4 3𝑥 𝑥 2 2𝑥 Exercises 45 4 Find the quotient and reduce to lowest terms 𝑥2 5𝑥 6 𝑥 4 𝑥2 9 𝑥2 5𝑥 4 5 Simplify the rational expression 𝑥 2 𝑥2 2𝑥 3 𝑥2 𝑥 6 𝑥2 6𝑥 5 2 4 X x c 2 3 3x V A 2x x 4 a Coox 3xl x 2 326x x 5x 6 x 5x 4 x 5x 4x 5x 25x 20x 6x 30x 24 x 4 x 9 x 9x 4x2 36 2 2 5x 5x23 4x2 25x2 6 20x 30x 24 x 15x 10x 24 3 3 x 4x2 9x 36 x 4x 9x 36 x 2 x 6x 5 jx x 2x 3 x x 6 x 6x 5x 2x 12x 10 x x 6x 2x 2x 12x 3x2 3x 18 x 8x 17x 10 x 3x 9x 17x 18 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S122 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Perform the following operations a Multiply 1 3 𝑥 2 by 9 b Divide 1 4 𝑥 8 by 1 12 c Multiply 1 4 1 3 𝑥 2 by 12 d Divide 1 3 2 5 𝑥 1 5 by 1 15 e Multiply 2 3 2𝑥 2 3 by 9 4 f Multiply 0034 𝑥 by 100 2 Write each rational expression as an equivalent rational expression in lowest terms a 𝑎3𝑏2 𝑐2𝑑2 𝑐 𝑎𝑏 𝑎 𝑐2𝑑3 b 𝑎26𝑎9 𝑎29 3𝑎9 𝑎3 c 6𝑥 4𝑥16 4𝑥 𝑥216 d 3𝑥26𝑥 3𝑥1 𝑥3𝑥2 𝑥24𝑥4 e 2𝑥210𝑥12 𝑥24 2𝑥 3𝑥 f 𝑎2𝑏 𝑎2𝑏 4𝑏2 𝑎2 g 𝑑𝑐 𝑐2𝑑2 𝑐2𝑑2 𝑑2𝑑𝑐 h 12𝑎27𝑎𝑏𝑏2 9𝑎2𝑏2 16𝑎2𝑏2 3𝑎𝑏𝑏2 i 𝑥3 𝑥24 1 𝑥2𝑥6 𝑥2 j 𝑥2 𝑥21 3 𝑥24𝑥4 𝑥22𝑥3 k 6𝑥211𝑥10 6𝑥25𝑥6 64𝑥 2520𝑥4𝑥2 l 3𝑥33𝑎2𝑥 𝑥22𝑎𝑥𝑎2 𝑎𝑥 𝑎3𝑥𝑎2𝑥2 Lesson Summary In this lesson we extended multiplication and division of rational numbers to multiplication and division of rational expressions To multiply two rational expressions multiply the numerators together and multiply the denominators together and then reduce to lowest terms To divide one rational expression by another multiply the first by the multiplicative inverse of the second and reduce to lowest terms To simplify a complex fraction apply the process for dividing one rational expression by another 1 a 9 zx 2 x 2 3x 2 3x 6 3 b 1 x 81 I 4 12 8 x 81 12 12x 96 3x 24 Y 92 47 197 4 ch I x 1 4 1 f f ob 1x 2 x b x Y dl 1 15 15 5x 2x 1 el 22x 2 es 3 Y 18 2x 2 36x 36 3x 1 12 12 36 Fl 0 03 4x 100 34 x 12 3x 2 2 2ad ab 2b cora ad abc2 2 b a 62 9 3a 9 22 9 a 3 3a 1822 27a ha 54a 31 32 922 27a 8 a 2a 322 27 2 322 92 27 c6 3 be 3x 48x 3xt 48 4x 16x264x Ext 32x 8x 32 d 3x2 6x 3x 2 3x 1 x2 4x 4 3x 9x 6x 18x 9x 15x bx 3 12x 12x x2 4x 4 3x 11322 8x 4 e 2x 10c 12 2 x c 4 3 x Yx 20x 27 2x 0x 12x 2x bx 8x 24 3x2 12 x 4x x 3x 4x 12 f a 2b 14b 24 2 2b 2 2b 2 2b 2 4ab2 2 8b2atb 2 2 1462a g d co d2 dc d C d dc ddc dc dc de dc2d C cd2 cd2d h 12 a 7ab ba 16abe 9yt 12 3ab 12 12a2 Zab b2 Sab ba 36ab 12ab21ab7ab 3 ab bu 92 Da 162 ba 1442 922 1622 D 362b aab2 42b bu 442 25a22 by S 2 il x 3 c x 6 3 x2 4 x 2 Y 3 x2 4 2 x 6 x x 6x 4x2 4x 24 x x 10x 4x 24 x 3 x 2 x2 2x 3x 6 x2 5x 6 3 j x 2 x 4x 4 2 2 1 x 2 2x 3 3 x 1 x 2x 3 x 2 2 4x 4 x x 2 1xt 1 x 2x 3 2 x22x 2x 4x 2 x 4x 4 xx x x2 xxx 1 2x 3 x 2x 2x Yx 2x 4x 4x 8x2 4x 4 4x 3x 3x2 1 x2x 3 x 6x 4x 8 x2 4x 4 x 2x 3x 3x6x 9x 3x 6x 9x2 x 2x 3 x 4x 4x bx 24x 24x2 4x 16x2 16x 8 32x 32 x 2 Ge c 6x bx Ex 2x 3 5 x Sox 24x 16x2 16x 32 al 6x 11x 10 6 4x 6x 5x 6 2520x 4x 36022 24x 66x 44x2 60 40x 150x2120x 24x 125x 10020 150 120x 24xt 24x 80x26x 60 2 x 140x 226x2 5x 150 I 3x 3a 2 x x 2ax 22 a a Bax 3x Sa 32 a a 2aLax a a Y 4 32x 32 3ax 3 L 3 5 ax 2x abx a4 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S123 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Write each rational expression as an equivalent rational expression in lowest terms a 4𝑎 6𝑏2 20𝑎3 12𝑏 b 𝑥2 𝑥21 𝑥24 𝑥6 c 𝑥22𝑥3 𝑥23𝑥4 𝑥2𝑥6 𝑥4 4 Suppose that 𝑥 𝑡23𝑡4 3𝑡23 and 𝑦 𝑡22𝑡8 2𝑡22𝑡4 for 𝑡 1 𝑡 1 𝑡 2 and 𝑡 4 Show that the value of 𝑥2𝑦2 does not depend on the value of 𝑡 5 Determine which of the following numbers is larger without using a calculator 1516 1615 or 2024 2420 Hint We can compare two positive quantities 𝑎 and 𝑏 by computing the quotient 𝑎 𝑏 If 𝑎 𝑏 1 then 𝑎 𝑏 Likewise if 0 𝑎 𝑏 1 then 𝑎 𝑏 Extension 6 One of two numbers can be represented by the rational expression 𝑥2 𝑥 where 𝑥 0 and 𝑥 2 a Find a representation of the second number if the product of the two numbers is 1 b Find a representation of the second number if the product of the two numbers is 0 42 12D 48ats 662 2023 1 120 a3 b2 x 2 x 6 x x 2x 12 xt8x 12 b 21 224 x 4x2 x 4 x 5x 4 2x 3 4 x 4x 2x7 8x 3x 12 3x 4 x x b x x 6x3x 3x2 18x 4x 4x 24 x 6x2 1x 12 x 4x3 7x2 22x 24 16 15 I 2024 CI 1615 2420 al x C y 1 y x x x 2 b c22 3 y 0 ift to y 4 x 72 3t 4 y ta 2t 8 3t2 3 2t2 2t 4 ag 2 x t 4t 1 t 4 x t 4 t 41 3t 1t 1 3t 1 3t 1 t 12 y t yt 2 t 4 y2 2t s yt 22 2t 2t 1 2t 1 t 4 t 412 xy 4 9t 112 x y2 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S27 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties Classwork Exercise 1 Suzy draws the following picture to represent the sum 3 4 Ben looks at this picture from the opposite side of the table and says You drew 4 3 Explain why Ben might interpret the picture this way Exercise 2 Suzy adds more to her picture and says The picture now represents 3 4 2 How might Ben interpret this picture Explain your reasoning Ben is looking from the opposite side like that So he sees it on the opposite order Ben Suzy too 2 4 3 Ben is looking from the opposite side like that So he sees it on ⑭ the opposite order Suzy Ben too NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S28 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 3 Suzy then draws another picture of squares to represent the product 3 4 Ben moves to the end of the table and says From my new seat your picture looks like the product 4 3 What picture might Suzy have drawn Why would Ben see it differently from his viewpoint Exercise 4 Draw a picture to represent the quantity 3 4 5 that also could represent the quantity 4 5 3 when seen from a different viewpoint Four Properties of Arithmetic THE COMMUTATIVE PROPERTY OF ADDITION If 𝑎 and 𝑏 are real numbers then 𝑎 𝑏 𝑏 𝑎 THE ASSOCIATIVE PROPERTY OF ADDITION If 𝑎 𝑏 and 𝑐 are real numbers then 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 THE COMMUTATIVE PROPERTY OF MULTIPLICATION If 𝑎 and 𝑏 are real numbers then 𝑎 𝑏 𝑏 𝑎 THE ASSOCIATIVE PROPERTY OF MULTIPLICATION If 𝑎 𝑏 and 𝑐 are real numbers then 𝑎𝑏𝑐 𝑎𝑏𝑐 Exercise 5 Viewing the diagram below from two different perspectives illustrates that 3 4 2 equals 2 4 3 Is it true for all real numbers 𝑥 𝑦 and 𝑧 that 𝑥 𝑦 𝑧 should equal 𝑧 𝑦 𝑥 Note The direct application of the associative property of addition only gives 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 Suzy Ben Yes NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S29 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 6 Draw a flow diagram and use it to prove that 𝑥𝑦𝑧 𝑧𝑦𝑥 for all real numbers 𝑥 𝑦 and 𝑧 Exercise 7 Use these abbreviations for the properties of real numbers and complete the flow diagram 𝐶 for the commutative property of addition 𝐶 for the commutative property of multiplication 𝐴 for the associative property of addition 𝐴 for the associative property of multiplication O xy z A xyz yx t zyk 2x Cx C C C NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S30 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 8 Let 𝑎 𝑏 𝑐 and 𝑑 be real numbers Fill in the missing term of the following diagram to show that 𝑎 𝑏 𝑐 𝑑 is sure to equal 𝑎 𝑏 𝑐 𝑑 NUMERICAL SYMBOL A numerical symbol is a symbol that represents a specific number For example 0 1 2 3 2 3 3 124122 𝜋 𝑒 are numerical symbols used to represent specific points on the real number line VARIABLE SYMBOL A variable symbol is a symbol that is a placeholder for a number It is possible that a question may restrict the type of number that a placeholder might permit eg integers only or positive real numbers 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 Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the commutative associative and distributive properties and the properties of rational exponents to components of the first expression NUMERICAL EXPRESSION A numerical expression is an algebraic expression that contains only numerical symbols no variable symbols which evaluate to a single number The expression 3 0 is not a numerical expression EQUIVALENT NUMERICAL EXPRESSIONS Two numerical expressions are equivalent if they evaluate to the same number Note that 1 2 3 and 1 2 3 for example are equivalent numerical expressions they are both 6 but 𝑎 𝑏 𝑐 and 𝑎 𝑏 𝑐 are not equivalent expressions allb c d NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S31 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 The following portion of a flow diagram shows that the expression 𝑎𝑏 𝑐𝑑 is equivalent to the expression 𝑑𝑐 𝑏𝑎 Fill in each circle with the appropriate symbol Either 𝐶 for the commutative property of addition or 𝐶 for the commutative property of multiplication 2 Fill in the blanks of this proof showing that 𝑤 5𝑤 2 is equivalent to 𝑤2 7𝑤 10 Write either commutative property associative property or distributive property in each blank 𝑤 5𝑤 2 𝑤 5𝑤 𝑤 5 2 𝑤𝑤 5 𝑤 5 2 𝑤𝑤 5 2𝑤 5 𝑤2 𝑤 5 2𝑤 5 𝑤2 5𝑤 2𝑤 5 𝑤2 5𝑤 2𝑤 10 𝑤2 5𝑤 2𝑤 10 𝑤2 7𝑤 10 Lesson Summary The commutative and associative properties represent key beliefs about the arithmetic of real numbers These properties can be applied to algebraic expressions using variables that represent real numbers Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the commutative associative and distributive properties and the properties of rational exponents to components of the first expression C Cx Ex Commutative multiplication Commutative multiplication Distributive multiplication Associative multiplication Distributive multiplication Commutative addition Associative addition NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S32 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Fill in each circle of the following flow diagram with one of the letters C for commutative property for either addition or multiplication A for associative property for either addition or multiplication or D for distributive property 4 What is a quick way to see that the value of the sum 53 18 47 82 is 200 5 a If 𝑎𝑏 37 and 1 37 what is the value of the product 𝑥 𝑏 𝑦 𝑎 b Give some indication as to how you used the commutative and associative properties of multiplication to evaluate 𝑥 𝑏 𝑦 𝑎 in part a c Did you use the associative and commutative properties of addition to answer Question 4 6 The following is a proof of the algebraic equivalency of 2𝑥3 and 8𝑥3 Fill in each of the blanks with either the statement commutative property or associative property 2𝑥3 2𝑥 2𝑥 2𝑥 2𝑥 2𝑥 2𝑥 22𝑥2𝑥𝑥 2 2𝑥 2𝑥 𝑥 2 22𝑥𝑥 𝑥 2 2 2𝑥 𝑥 𝑥 8𝑥3 C C D C C C A D D C C 53 47 100 18782 100 100 100 200 37 1 x b y a y b 2 137 5 Yes Associative commutative Associative commutative Associative NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 7 ALGEBRA I Lesson 7 Algebraic ExpressionsThe Commutative and Associative Properties S33 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 7 Write a mathematical proof of the algebraic equivalency of 𝑎𝑏2 and 𝑎2𝑏 2 8 a Suppose we are to play the 4number game with the symbols 𝑎 𝑏 𝑐 and 𝑑 to represent numbers each used at most once combined by the operation of addition ONLY If we acknowledge that parentheses are unneeded show there are essentially only 15 expressions one can write b How many answers are there for the multiplication ONLY version of this game 9 Write a mathematical proof to show that 𝑥 𝑎𝑥 𝑏 is equivalent to 𝑥2 𝑎𝑥 𝑏𝑥 𝑎𝑏 10 Recall the following rules of exponents 𝑥𝑎 𝑥𝑏 𝑥𝑎𝑏 𝑥𝑎 𝑥𝑏 𝑥𝑎𝑏 𝑥𝑎𝑏 𝑥𝑎𝑏 𝑥𝑦𝑎 𝑥𝑎𝑦𝑎 𝑥 𝑦 𝑎 𝑥𝑎 𝑦𝑎 Here 𝑥 𝑦 𝑎 and 𝑏 are real numbers with 𝑥 and 𝑦 nonzero Replace each of the following expressions with an equivalent expression in which the variable of the expression appears only once with a positive number for its exponent For example 7 𝑏2 𝑏4 is equivalent to 7 𝑏6 a 16𝑥2 16𝑥5 b 2𝑥42𝑥3 c 9𝑧23𝑧13 d 25𝑤4 5𝑤3 5𝑤7 e 25𝑤4 5𝑤3 5𝑤7 Optional Challenge 11 Grizelda has invented a new operation that she calls the average operator For any two real numbers 𝑎 and 𝑏 she declares 𝑎 𝑏 to be the average of 𝑎 and 𝑏 𝑎 𝑏 𝑎 𝑏 2 a Does the average operator satisfy a commutative property That is does 𝑎 𝑏 𝑏 𝑎 for all real numbers 𝑎 and 𝑏 b Does the average operator distribute over addition That is does 𝑎𝑏 𝑐 𝑎𝑏 𝑎𝑐 for all real numbers 𝑎 𝑏 and 𝑐 labl lablabl 2 b a b a 2 b b 22b2 x 4 bx 2x ab 8 ala b 2 d Sterm Yoptions a b a c a dib c b d c d 2 terms 6 options 2 b ca b d b c d a c d Sterms 4 options a b c d Y terms I option 4 6 4 1 15 b 2 b c d Y options 2 b a cia dib c b d cd6 options a b c 2 bid a cd b cdo Y options a D c d o I option 4 6 y 1 25 10 al 16x2 x 3 b 2x4 3 2x1 16x5 I I 2 192 37437 3z1z 3 al 252 5rS 25 5wS S 5w el 25w 25 y Y 25 b Jus wso 25w 5w NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S119 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 24 Multiplying and Dividing Rational Expressions Classwork Example 1 Make a conjecture about the product 𝑥3 4𝑦 𝑦2 𝑥 What will it be Explain your conjecture and give evidence that it is correct Example 2 Find the following product 3𝑥 6 2𝑥 6 5𝑥 15 4𝑥 8 𝑎 𝑏 𝑐 𝑑 𝑎𝑐 𝑏𝑑 If 𝑎 𝑏 𝑐 and 𝑑 are rational expressions with 𝑏 0 𝑑 0 then 3 2 the process of multiplying and r y sy ray simplifying shows that this expression 4y x 4 is correct Each step involved using algebric rules such as multiplying fractions canceling common terms and simplifying the result which confirms that the Final expression is valid 15x2 45x 30x 90 8x 4 16x 24x 48 15x2 15x 90 8x2 40x 48 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S120 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercises 13 1 Summarize what you have learned so far with your neighbor 2 Find the following product and reduce to lowest terms 2𝑥6 𝑥2𝑥6 𝑥24 2𝑥 3 Find the following product and reduce to lowest terms 4𝑛12 3𝑚6 2 𝑛22𝑛3 𝑚24𝑚4 𝑎 𝑏 𝑐 𝑑 𝑎 𝑏 𝑑 𝑐 𝑎𝑑 𝑏𝑐 If 𝑎 𝑏 𝑐 and 𝑑 are rational expressions with 𝑏 0 𝑐 0 and 𝑑 0 then 2x 8x 6x2 24 2x 2x 12x 2x 6x Ex24 2x 2x2 12x 11 Eme 2 3m3m b 9m 18m 18m 36 X 4n 12 Yn 12 Yn 12 1n2 48n 48u 144 9m2 36m 36 n 2 2n 3 Emen 18mn27m 36mn72mn108m 36n72n108 I 16n2 96n 144 ma ym 4 1 nim 64ndm 64na 964m2 384nm 38yn 144m2 576m 576 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S121 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Example 3 Find the quotient and reduce to lowest terms 𝑥2 4 3𝑥 𝑥 2 2𝑥 Exercises 45 4 Find the quotient and reduce to lowest terms 𝑥2 5𝑥 6 𝑥 4 𝑥2 9 𝑥2 5𝑥 4 5 Simplify the rational expression 𝑥 2 𝑥2 2𝑥 3 𝑥2 𝑥 6 𝑥2 6𝑥 5 2 4 X x c 2 3 3x V A 2x x 4 a Coox 3xl x 2 326x x 5x 6 x 5x 4 x 5x 4x 5x 25x 20x 6x 30x 24 x 4 x 9 x 9x 4x2 36 2 2 5x 5x23 4x2 25x2 6 20x 30x 24 x 15x 10x 24 3 3 x 4x2 9x 36 x 4x 9x 36 x 2 x 6x 5 jx x 2x 3 x x 6 x 6x 5x 2x 12x 10 x x 6x 2x 2x 12x 3x2 3x 18 x 8x 17x 10 x 3x 9x 17x 18 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S122 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Perform the following operations a Multiply 1 3 𝑥 2 by 9 b Divide 1 4 𝑥 8 by 1 12 c Multiply 1 4 1 3 𝑥 2 by 12 d Divide 1 3 2 5 𝑥 1 5 by 1 15 e Multiply 2 3 2𝑥 2 3 by 9 4 f Multiply 0034 𝑥 by 100 2 Write each rational expression as an equivalent rational expression in lowest terms a 𝑎3𝑏2 𝑐2𝑑2 𝑐 𝑎𝑏 𝑎 𝑐2𝑑3 b 𝑎26𝑎9 𝑎29 3𝑎9 𝑎3 c 6𝑥 4𝑥16 4𝑥 𝑥216 d 3𝑥26𝑥 3𝑥1 𝑥3𝑥2 𝑥24𝑥4 e 2𝑥210𝑥12 𝑥24 2𝑥 3𝑥 f 𝑎2𝑏 𝑎2𝑏 4𝑏2 𝑎2 g 𝑑𝑐 𝑐2𝑑2 𝑐2𝑑2 𝑑2𝑑𝑐 h 12𝑎27𝑎𝑏𝑏2 9𝑎2𝑏2 16𝑎2𝑏2 3𝑎𝑏𝑏2 i 𝑥3 𝑥24 1 𝑥2𝑥6 𝑥2 j 𝑥2 𝑥21 3 𝑥24𝑥4 𝑥22𝑥3 k 6𝑥211𝑥10 6𝑥25𝑥6 64𝑥 2520𝑥4𝑥2 l 3𝑥33𝑎2𝑥 𝑥22𝑎𝑥𝑎2 𝑎𝑥 𝑎3𝑥𝑎2𝑥2 Lesson Summary In this lesson we extended multiplication and division of rational numbers to multiplication and division of rational expressions To multiply two rational expressions multiply the numerators together and multiply the denominators together and then reduce to lowest terms To divide one rational expression by another multiply the first by the multiplicative inverse of the second and reduce to lowest terms To simplify a complex fraction apply the process for dividing one rational expression by another 1 a 9 zx 2 x 2 3x 2 3x 6 3 b 1 x 81 I 4 12 8 x 81 12 12x 96 3x 24 Y 92 47 197 4 ch I x 1 4 1 f f ob 1x 2 x b x Y dl 1 15 15 5x 2x 1 el 22x 2 es 3 Y 18 2x 2 36x 36 3x 1 12 12 36 Fl 0 03 4x 100 34 x 12 3x 2 2 2ad ab 2b cora ad abc2 2 b a 62 9 3a 9 22 9 a 3 3a 1822 27a ha 54a 31 32 922 27a 8 a 2a 322 27 2 322 92 27 c6 3 be 3x 48x 3xt 48 4x 16x264x Ext 32x 8x 32 d 3x2 6x 3x 2 3x 1 x2 4x 4 3x 9x 6x 18x 9x 15x bx 3 12x 12x x2 4x 4 3x 11322 8x 4 e 2x 10c 12 2 x c 4 3 x Yx 20x 27 2x 0x 12x 2x bx 8x 24 3x2 12 x 4x x 3x 4x 12 f a 2b 14b 24 2 2b 2 2b 2 2b 2 4ab2 2 8b2atb 2 2 1462a g d co d2 dc d C d dc ddc dc dc de dc2d C cd2 cd2d h 12 a 7ab ba 16abe 9yt 12 3ab 12 12a2 Zab b2 Sab ba 36ab 12ab21ab7ab 3 ab bu 92 Da 162 ba 1442 922 1622 D 362b aab2 42b bu 442 25a22 by S 2 il x 3 c x 6 3 x2 4 x 2 Y 3 x2 4 2 x 6 x x 6x 4x2 4x 24 x x 10x 4x 24 x 3 x 2 x2 2x 3x 6 x2 5x 6 3 j x 2 x 4x 4 2 2 1 x 2 2x 3 3 x 1 x 2x 3 x 2 2 4x 4 x x 2 1xt 1 x 2x 3 2 x22x 2x 4x 2 x 4x 4 xx x x2 xxx 1 2x 3 x 2x 2x Yx 2x 4x 4x 8x2 4x 4 4x 3x 3x2 1 x2x 3 x 6x 4x 8 x2 4x 4 x 2x 3x 3x6x 9x 3x 6x 9x2 x 2x 3 x 4x 4x bx 24x 24x2 4x 16x2 16x 8 32x 32 x 2 Ge c 6x bx Ex 2x 3 5 x Sox 24x 16x2 16x 32 al 6x 11x 10 6 4x 6x 5x 6 2520x 4x 36022 24x 66x 44x2 60 40x 150x2120x 24x 125x 10020 150 120x 24xt 24x 80x26x 60 2 x 140x 226x2 5x 150 I 3x 3a 2 x x 2ax 22 a a Bax 3x Sa 32 a a 2aLax a a Y 4 32x 32 3ax 3 L 3 5 ax 2x abx a4 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 24 ALGEBRA II Lesson 24 Multiplying and Dividing Rational Expressions S123 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Write each rational expression as an equivalent rational expression in lowest terms a 4𝑎 6𝑏2 20𝑎3 12𝑏 b 𝑥2 𝑥21 𝑥24 𝑥6 c 𝑥22𝑥3 𝑥23𝑥4 𝑥2𝑥6 𝑥4 4 Suppose that 𝑥 𝑡23𝑡4 3𝑡23 and 𝑦 𝑡22𝑡8 2𝑡22𝑡4 for 𝑡 1 𝑡 1 𝑡 2 and 𝑡 4 Show that the value of 𝑥2𝑦2 does not depend on the value of 𝑡 5 Determine which of the following numbers is larger without using a calculator 1516 1615 or 2024 2420 Hint We can compare two positive quantities 𝑎 and 𝑏 by computing the quotient 𝑎 𝑏 If 𝑎 𝑏 1 then 𝑎 𝑏 Likewise if 0 𝑎 𝑏 1 then 𝑎 𝑏 Extension 6 One of two numbers can be represented by the rational expression 𝑥2 𝑥 where 𝑥 0 and 𝑥 2 a Find a representation of the second number if the product of the two numbers is 1 b Find a representation of the second number if the product of the two numbers is 0 42 12D 48ats 662 2023 1 120 a3 b2 x 2 x 6 x x 2x 12 xt8x 12 b 21 224 x 4x2 x 4 x 5x 4 2x 3 4 x 4x 2x7 8x 3x 12 3x 4 x x b x x 6x3x 3x2 18x 4x 4x 24 x 6x2 1x 12 x 4x3 7x2 22x 24 16 15 I 2024 CI 1615 2420 al x C y 1 y x x x 2 b c22 3 y 0 ift to y 4 x 72 3t 4 y ta 2t 8 3t2 3 2t2 2t 4 ag 2 x t 4t 1 t 4 x t 4 t 41 3t 1t 1 3t 1 3t 1 t 12 y t yt 2 t 4 y2 2t s yt 22 2t 2t 1 2t 1 t 4 t 412 xy 4 9t 112 x y2