Lesson 6: Algebraic Expressions—The Distributive Property

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1 ALGEBRA I

Classwork Exercise 2 Write an expression using the symbols 1, 2, 3, and 4 that evaluates to 16, and then use that expression to create one that evaluates to 816.

Exercise 3 Define the rules of a game as follows: a. Begin by choosing an initial set of symbols, variable, or numeric, as a starting set of expressions, and then generate more expressions by:

b. Placing any previously created expressions into the blanks of the addition operator: ___+ ___

Exercise 4 Roma says that collecting like terms can be seen as an application of the distributive property. Is writing + = 2 an application of the distributive property? 푥 푥 푥

Lesson 6: Algebraic Expressions—The Distributive Property Date: 8/7/13 S.26

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1 ALGEBRA I

Exercise 5 Leela is convinced that ( + ) = + . Do you think she is right? Use a picture to illustrate your reasoning. 2 2 2 푎 푏 푎 푏

Exercise 6 Draw a picture to represent the expression ( + + 1) × ( + 1).

푎 푏 푏

Exercise 7 Draw a picture to represent the expression ( + ) × ( + ) × ( + + ).

푎 푏 푐 푑 푒 푓 푔

A Key Belief of Arithmetic The Distributive Property: If , , and are real numbers, then ( + ) = + .

푎 푏 푐 푎 푏 푐 푎푏 푎푐

Lesson 6: Algebraic Expressions—The Distributive Property Date: 8/7/13 S.27

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1 ALGEBRA I

Lesson Summary The Distributive Property represents a key belief about the arithmetic of real numbers. This property can be applied to algebraic expressions using variables that represent real numbers.

Problem Set

1. Insert parentheses to make each statement true. a. 2 + 3 × 4 + 1 = 81 b. 2 + 3 × 42 + 1 = 85 c. 2 + 3 × 42 + 1 = 51 d. 2 + 3 × 42 + 1 = 53 2 2. Using starting symbols of: , , 2, and – 2, which of the following expressions will NOT appear when following the rules of the game played in Exercise 3? 푤 푞 a. 7 + 3 + ( 2) b. 2 푤 푞 − c. 푞 − d. 2 + 6 푤 − 푞 e. 2 + 2 푤 − 푤 3. Luke wants to play the 4-number game with the numbers 1, 2, 3, and 4 and the operations of addition, multiplication, AND subtraction. Leoni responds, “Or we just could play the 4-number game with just the operations of addition and multiplication, but with now the numbers –1, –2, –3, –4, 1, 2, 3, and 4 instead.” What observation is Leoni trying to point out to Luke?

4. Given the expression: ( + 3) ( + 1) ( + 2) a. Draw a picture to represent the expression. 푥 ∙ 푦 ∙ 푥 b. Write an equivalent expression by applying the Distributive Property.

Lesson 6: Algebraic Expressions—The Distributive Property Date: 8/7/13 S.28

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1 ALGEBRA I

5. a. Given that > , which of the shaded regions is larger and why?

푎 푏

b. Consider the expressions 851 × 29 and 849 × 31. Which would result in a larger product? Use a diagram to demonstrate your result

6. Consider the following diagram.

Edna looked at the diagram and then highlighted the four small rectangles shown and concluded: ( + 2 ) = + 4 ( + ). 2 2 푥 푎 푥 푎 푥 푎

Lesson 6: Algebraic Expressions—The Distributive Property Date: 8/7/13 S.29

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1 ALGEBRA I

a. Michael, when he saw the picture, highlighted four rectangles and concluded: ( + 2 ) = + 2 + 2 ( + 2 ). Which four2 rectangles2 and one square did he highlight? 푥 푎 푥 푎푥 푎 푥 푎

b. Jill, when she saw the picture, highlighted eight rectangles and squares (not including the square in the middle) to conclude: ( + 2 ) = + 4 + 4 . Which eight2 rectangles2 and squares2 did she highlight? 푥 푎 푥 푎푥 푎

c. When Fatima saw the picture, she exclaimed: ( + 2 ) = + 4 ( + 2 ) 4 . She claims she highlighted just four rectangles to conclude this. Identify2 the 2four rectangles she highlig2 hted and explain how using them she arrived that the expression +푥 4 (푎 + 2 푥) 4 푎 . 푥 푎 − 푎 2 2 푥 푎 푥 푎 − 푎 d. Are all of Edna, Michael, Jill, and Fatima’s techniques right? Explain why or why not?

Lesson 6: Algebraic Expressions—The Distributive Property Date: 8/7/13 S.30

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 ALGEBRA I

Lesson 7: Algebraic Expressions—The 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.

Lesson 7: Algebraic Expressions—The Commutative and Associative Properties Date: 8/9/13 S.31

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 ALGEBRA I

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 and are real numbers, then + = + .

The Associative Property of Addition: If , 푎, and 푏 are real numbers, then 푎( +푏 ) +푏 =푎 + ( + ) The Commutative Property of Multiplication:푎 푏 If 푐 and are real numbers, then푎 푏 × 푐= 푎× . 푏 푐 The Associative Property of Multiplication: If , 푎, and 푏 are real numbers, then 푎( )푏 =푏 ( 푎). 푎 푏 푐 푎푏 푐 푎 푏푐 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 , and that ( + ) + should equal ( + ) + ?

(Note: The direct application푥 of푦 the Associative푧 푥 Property푦 푧 of Addition only푧 gives푦 ( 푥+ ) + = + ( + ).) 푥 푦 푧 푥 푦 푧

Lesson 7: Algebraic Expressions—The Commutative and Associative Properties Date: 8/9/13 S.32

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 ALGEBRA I

Exercise 6 Draw a flow diagram and use it to prove that ( ) = ( ) for all real numbers , , and .

푥푦 푧 푧푦 푥 푥 푦 푧

Exercise 7 Use these abbreviations for the properties of real numbers and complete the flow diagram. for the commutative property of addition

퐶×+ for the commutative property of multiplication 퐶 for the associative property of addition

퐴×+ for the associative property of multiplication 퐴

Lesson 7: Algebraic Expressions—The Commutative and Associative Properties Date: 8/9/13 S.33

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 ALGEBRA I

Exercise 8 Let , , , and be real numbers. Fill in the missing term of the following diagram to show that ( + ) + + is sure to equal + + ( + ) . 푎 푏 푐 푑 � 푎 푏 푐� 푑 푎 �푏 푐 푑 �

NUMERICAL SYMBOL: A numerical symbol is a symbol that represents a specific number. 2 0, 1, 2, 3, , 3, 124.122, , For example, 3 are numerical symbols used to represent specific points on the real number line. − − 휋 푒 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, e.g., integers only or positive real number. 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 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 + + and × × are not equivalent expressions. 푎 푏 푐 푎 푏 푐

Lesson 7: Algebraic Expressions—The Commutative and Associative Properties Date: 8/9/13 S.34

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 ALGEBRA I

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.

Problem Set

1. The following portion of a flow diagram shows that the expression + is equivalent to the expression + .

풂풃 풄풅 풅풄 풃풂

Fill in each circle with the appropriate symbol: Either (for the “Commutative Property of Addition”) or × (for the “Commutative Property of Multiplication”). 퐶+ 퐶

2. Fill in the blanks of this proof showing that ( + 5)( + 2) is equivalent + 7 + 10. Write either “Commutative Property,” “Associative Property,” or “Distributive Property”2 in each blank. 푤 푤 푤 푤

( + 5)( + 2) = ( + 5) + ( + 5) × 2 = ( + 5) + ( + 5) × 2 푤 푤 = 푤( + 5푤) + 2(푤 + 5) = 푤 푤+ × 5 +푤 2( + 5) = 푤2푤+ 5 + 2( 푤+ 5) = 푤2 + 5푤 + 2 +푤10 = 푤2 + (5푤 + 2푤) + 10 = 푤2 + 7푤 + 10푤 푤2 푤 푤 푤 푤

Lesson 7: Algebraic Expressions—The Commutative and Associative Properties Date: 8/9/13 S.35

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 ALGEBRA I

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?

a. If = 37 and = , what is the value of the product × × × ? 1 b. Give some indication as to how you used the commutative and associative properties of multiplication to 푎푏 37 푥 푏 푦 푎 evaluate × × × in part a). c. Did you use the associative and commutative properties of addition to answer question 4? 푥 푏 푦 푎

6. The following is a proof of the algebraic equivalency of (2 ) and 8 . Fill in each of the blanks with either the statement “Commutative Property” or “Associative Property.”3 3 푥 푥

(2 ) = 2 2 2 = 23( × 2)( × 2) ______푥 푥 ∙ 푥 ∙ = 2(2 )(2 ) ______푥 푥 푥 = 2 2( × 2) ______푥 푥 푥 = 2 2(2 ) ______∙ 푥 푥 ∙ 푥 = (2 2 2)( ) ______∙ 푥 푥 ∙ 푥 = 8 ∙ ∙ 푥 ∙ 푥 ∙ 푥 3 푥

Lesson 7: Algebraic Expressions—The Commutative and Associative Properties Date: 8/9/13 S.36

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 ALGEBRA I

7. Write a mathematical proof of the algebraic equivalency of ( ) and . 2 2 2 푎푏 푎 푏 8. a. Suppose we are to play the 4-number 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 ( + )( + ) is equivalent to + + + . 2 푥 푎 푥 푏 푥 푎푥 푏푥 푎푏 10. Recall the following rules of exponents:

= = ( ) = 푎 푎 푏 푎+푏 푥 푎−푏 푎 푏 푎푏 푥 ∙ 푥 푥 푏 푥 푥 푥 ( ) = = 푥 푎 푎 푎 푎 푎 푥 푥 푥푦 푥 푦 � � 푎 푦 푦 Here , , , and are real numbers with and non-zero.

Replace푥 each푦 푎 of the푏 following expressions with푥 an푦 equivalent expression in which the variable of the expression appears only once with a positive number for its exponent. (For example, is equivalent to .) 7 −4 7 2 6 푏 ∙ 푏 푏 a. (16 ) ÷ (16 ) b. (2 ) 2(2 ) 5 푥 푥 c. (9 4 )(3 3 ) 푥 푥 d. (25−2 ) ÷−1(5−3 ) ÷ (5 ) 푧 푧 (25 4) ÷ (5 3) ÷ (5 −7) e. � 푤 푤 � 푤 4 3 −7 푤 � 푤 푤 � 11. (OPTIONAL CHALLENGE) Grizelda has invented a new operation that she calls the “average operator.” For any two real numbers and , she declares to be the average of and : + 푎 푏 푎 ⨁ 푏 = 푎 푏 2 푎 푏 a. Does the average operator satisfy a commutative푎 ⨁ 푏 property? That is, does = for all real numbers and ? 푎 ⨁ 푏 푏 ⨁ 푎 b. Does the average operate distribute over addition? That is, does ( + ) = ( ) + ( ) for all real 푎 푏 numbers , , and ? 푎⨁ 푏 푐 푎⨁푏 푎⨁푐 푎 푏 푐

Lesson 7: Algebraic Expressions—The Commutative and Associative Properties Date: 8/9/13 S.37

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1 ALGEBRA I

Lesson 8: Adding and Subtracting Polynomials

Classwork Exercise 1 a. How many quarters, nickels, and pennies are needed to make $1.13?

b. Fill in the blanks:

8,943 = ______× 1000 + ______× 100 + ______× 10 + ______× 1

= ______× 10 + ______× 10 + ______× 10 + ______× 1 3 2

c. Fill in the blanks:

8,943 = ______× 20 + ______× 20 + ______× 20 + ______× 1 3 2

d. Fill in the blanks:

113 = ______× 5 + ______× 5 + ______× 1 2

Exercise 2 Now let’s be as general as possible by not identifying which base we are in. Just call the base x. Consider the expression: 1 × + 2 × + 7 × + 3 × 1, or equivalently: + 2 + 7 + 3. 3 2 3 2 a. What is the value of this푥 expression푥 if =푥 10? 푥 푥 푥

푥

b. What is the value of this expression if = 20

푥

Lesson 8: Adding and Subtracting Polynomials Date: 8/7/13 S.38

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1 ALGEBRA I

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 22 + 3 when = 5? How much money is 22 nickels and 3 pennies?

푥 푥

c. What number is represented by 4 + 17 + 2 if = 10? 2 푥 푥 푥

d. What number is represented by 4 + 17 + 2 if = 2 or if = ? 2 2 푥 푥 푥 − 푥 3

e. What number is represented by 3 + 2 + when = 2? 2 1 − 푥 √ 푥 2 푥 √

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 (__×__).

Lesson 8: Adding and Subtracting Polynomials Date: 8/7/13 S.39

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1 ALGEBRA I

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. (4 + + 7) + (2 + 3 + 1). 2 2 푥 푥 푥 푥

c. (3 + 8) ( + 5 + 4 7). 3 2 3 2 푥 − 푥 − 푥 푥 푥 −

d. (3 + 8 ) 2( + 12). 3 3 푥 푥 − 푥

e. (5 ) + (9 + ) 2 2 − 푡 − 푡 푡 푡

f. (3 + 1) + 6( 8) ( + 2)

푝 푝 − − 푝

Lesson 8: Adding and Subtracting Polynomials Date: 8/7/13 S.40

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1 ALGEBRA I

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.

Problem Set

1. Celina says that each of the following expressions is actually a binomial in disguise: i. 5 2 + 6 ii. 5 2 2 10 + 3 + 3 ( 2) 푎푏푐 − 푎 푎푏푐 iii. ( +3 2) 2 4 4 5 4 푥 ∙ 푥 − 푥 푥 푥 ∙ − 푥 iv. 5( 12) 10( 1) + 100( 1) 푡 − 푡 v. (2 ) (2 ) 2 푎 − − 푎 − 푎 − 2 2 휋푟 − 휋푟 푟 − 휋푟 − 휋푟 ∙ 푟 For example, she sees that the expression in (i) is algebraically equivalent to 11 2 , which is indeed a binomial. (She is happy to write this as 11 + ( 2) , if you prefer.) 2 푎푏푐 − 푎 Is she right about the remaining four expressions? 2 푎푏푐 − 푎

2. Janie writes a polynomial expression using only one variable, , with degree 3. Max writes a polynomial expression using only one variable, , with degree 7. 푥 a. What can you determine about the degree of the sum of Janie and Max’s polynomials? 푥 b. What can you determine about the degree of the difference of Janie and Max’s polynomials?

3. Suppose Janie writes a polynomial expression using only one variable, , with degree of 5 and Max writes a polynomial expression using only one variable, , with degree of 5. 푥 a. What can you determine about the degree of the sum of Janie and Max’s polynomials? 푥 b. What can you determine about the degree of the difference of Janie and Max’s polynomials?

4. Find each sum or difference by combining the parts that are alike. a. (2 + 4) + 5( 1) ( + 7) b. (7 + 9 ) 2( + 13) 푝 푝 − − 푝 c. (6 4 ) + (9 4+ ) 푥 푥 − 푥 4 4 − 푡 − 푡 푡 푡

Lesson 8: Adding and Subtracting Polynomials Date: 8/7/13 S.41

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1 ALGEBRA I

d. (5 ) + 6( 8) ( + 12) e. (8 +2 5 ) 32( + 2) 2 − 푡 푡 − − 푡 f. (123 + 1) + 2( 3 4) ( 15) 푥 푥 − 푥 g. (13 + 5 ) 2( + 1) 푥 푥 − − 푥 − 2 3 2 (9 ) (8 + 2 ) h. 푥 푥 −2 푥 2 2 i. (4 −+푡 −6)푡 12− ( 푡 3) +푡 ( + 2) j. (15 + 10 ) 12( + 4 ) 푚 − 푚 − 푚 4 4 푥 푥 − 푥 푥

Lesson 8: Adding and Subtracting Polynomials Date: 8/7/13 S.42

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M1 ALGEBRA I

Lesson 9: Multiplying Polynomials

Classwork Exercise 1 a. Gisella computed 342 × 23 as follows:

Can you explain what she is doing? What is her final answer?

Use a geometric diagram to compute the following products:

b. (3 + 4 + 2) × (2 + 3) 2 푥 푥 푥

c. (2 + 10 + 1)( + + 1) 2 2 푥 푥 푥 푥

Lesson 9: Multiplying Polynomials Date: 8/9/13 S.43

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M1 ALGEBRA I

d. ( 1)( + 6 5) 3 2 푥 − 푥 푥 −

Exercise 2 Multiply the polynomials using the distributive property: (3 + 1)( 2 + 1) 2 4 푥 푥 − 푥 − 푥

Exercise 3 The expression 10 + 6 is the result of applying the distributive property to the expression 2 (5 + ). It is also the result of the applying2 the distributive3 property to 2(5 + 3 ) or to (10 + 6 ), for example, 2or even to 1 (10 + 6 )! 푥 푥 2 3 2 푥 푥 2 3 푥 푥 푥 푥 푥 ∙ 푥 For푥 (i) to (x) below, write down an expression such that if you applied the distributive property to your expression it will give the result presented. Give interesting answers! i. 6 + 14 2 푎 푎

ii. 2 + 2 + 2 4 5 10 푥 푥 푥

iii. 6 15 2 푧 − 푧

Lesson 9: Multiplying Polynomials Date: 8/9/13 S.44

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M1 ALGEBRA I

iv. 42 14 + 77 3 5 푤 − 푤 푤

v. ( + ) + ( + ) 2 3 푧 푎 푏 푧 푎 푏

vi. 2 + 3 1 2 푠 2

vii. 15 6 + 9 + 3 2 3 4 2 5 4 2 3 6 푝 푟 − 푝 푟 푝 푟 √ 푝 푟

viii. 0.4 40 9 8 푥 − 푥

ix. (4 + 3)( + ) (2 + 2)( + ) 2 3 2 3 푥 푥 푥 − 푥 푥 푥

x. (2 + 5)( 2) (13 26)( 3)

푧 푧 − − 푧 − 푧 −

Lesson 9: Multiplying Polynomials Date: 8/9/13 S.45

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M1 ALGEBRA I

Exercise 4 Sammy wrote a polynomial using only one variable, , of degree 3. Myisha wrote a polynomial in the same variable of degree 5. What can you say about the degree of the product of Sammy and Myisha’s polynomials? 푥

Extension Find a polynomial that, when multiplied by 2 + 3 + 1, gives the answer 2 + 2 1. 2 3 2 푥 푥 푥 푥 − 푥 −

Lesson 9: Multiplying Polynomials Date: 8/9/13 S.46

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M1 ALGEBRA I

Problem Set

1. Use the distributive property to write each of the following expressions as the sum of monomials. a. 3 (4 + )

b. 푎( + 2)푎+ 1

c. 푥 (푥12 + 18 2) 1 d. 43 ( 푧 10)푧 3 e. (푥 푥 4)(− + 5) f. (2푥 − 1)(3푥 + 1) 2 g. (10푧 − 1)(푧10 + 1) h. ( 5푤 − 3) 푤 1 2 16 + 0.125 i. − 푤 −2 푤 100 200 j. (2푠 + 1)(2� 푠 + 1) 푠� 2 k. ( 푞 + 1)(푞 1) 2 l. 3푥 (−9 푥 + ) 푥 −2 ( + ) m. (푥푧 1)(푥푦 +푧 1)(− 푦푧+ 1)푥 푦 − 푧 2 n. (푡 −+ 1)(푡 푡 + + 1) 4 3 2 o. (2푤 + 1)(3푤 − 푤2) 푤 − 푤 p. 푧 ( +푧 )( +푧 −)( + )

q. 푥 푦 푦 푧 푧 푥 푥+푦 r. (320 10 ) ÷ 5 10 5 s. 5 푓( −+ 푓 2) 2(2 ) ( 2)( ) 3 t. − 푦 푦 푦 − − − 푦 푎+푏−푐 푎+푏+푐 u. (2 ÷ 917 + (5 ) ÷ 2) ÷ ( 2) v. ( 푥2 2 +푥 1)( −+ 2) 3 2 − 푓 − 푓 푓 − 푓

Lesson 9: Multiplying Polynomials Date: 8/9/13 S.47

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M1 ALGEBRA I

2. Use the distributive property (and your wits!) to write each of the following expressions as a sum of monomials. If the resulting polynomial is in one variable, write the polynomial in standard form.

a. ( + ) b. ( + 1) c. (3 + ) 2 2 2 푎 푏 푎 푏 d. (3 + 1) e. ( + + ) f. ( + 1 + ) 2 2 2 푥 푦 푧 푥 푧 g. (3 + ) h. ( + ) i. ( 1) 2 3 3 푧 푝 푞 푝 − j. (5 + ) 3 푞

3. Use the distributive property (and your wits!) to write each of the following expressions as a polynomial in standard form.

a. ( + 4)( 1) 2 b. 3(푠 + 4)(푠 − 1) 2 c. (푠 + 4)(푠 − 1) 2 d. 푠( 푠+ 1)( 푠+ − 4)( 1) 2 e. (푠 1)(푠 + 푠+ − + + + 1) 5 4 3 2 f. 5푢( − 1)(푢 +푢 +푢 +푢 +푢 + 1) 5 4 3 2 g. √ ( 푢+ − +푢 1)( 푢 1)(푢 +푢 +푢 + + + 1) 7 3 5 4 3 2 푢 푢 푢 − 푢 푢 푢 푢 푢 4. Beatrice writes down every expression that appears in this problem set, one after the other, linking them with “+” signs between them. She is left with one very large expression on her page. Is that expression a polynomial expression? That is, is it algebraically equivalent to a polynomial?

What if she wrote “ – ” signs between the expressions instead?

What if she wrote “×” signs between the expressions instead?

Lesson 9: Multiplying Polynomials Date: 8/9/13 S.48