Lesson 1: Multiplying And Factoring Polynomial Expressions

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Lesson 1NYS COMMON CORE MATHEMATICS CURRICULUMM4ALGEBRA ILesson 1: Multiplying and Factoring Polynomial ExpressionsStudent Outcomes Students use the distributive property to multiply a monomial by a polynomial and understand that factoringreverses the multiplication process. Students use polynomial expressions as side lengths of polygons and find area by multiplying. Students recognize patterns and formulate shortcuts for writing the expanded form of binomials whoseexpanded form is a perfect square or the difference of perfect squares.Lesson NotesCentral to the concepts in this lesson is A-APR.A.1 and understanding the system and operations of polynomialexpressions, specifically multiplication and factoring of polynomials. Lengths of time suggested for the examples andexercises in this lesson assume that students remember some of what is presented in the examples from work in earliermodules and earlier grades. Students may need more or less time with this lesson than is suggested. The teacher shouldmake decisions about how much time is needed for these concepts based on studentsโ€™ needs.MP.4This lesson asks students to use geometric models to demonstrate their understanding of multiplication of polynomials.ClassworkOpening Exercise (4 minutes)Opening ExerciseWrite expressions for the areas of the two rectangles in the figures given below.๐’›๐Ÿ๐’›๐Ÿ–๐Ÿ๐’›Blue: ๐Ÿ๐’›(๐’›) ๐Ÿ๐’›๐Ÿ; Orange: ๐Ÿ๐’›(๐Ÿ–) ๐Ÿ๐Ÿ”๐’›Lesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201517This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Lesson 1NYS COMMON CORE MATHEMATICS CURRICULUMM4ALGEBRA INow write an expression for the area of this rectangle:๐’›๐Ÿ–๐Ÿ๐’›๐Ÿ๐’›๐Ÿ ๐Ÿ๐Ÿ”๐’› How did you find your answer for the second rectangle? If you find the area by multiplying the total length times total width, what property of operations are youusing? Add the two areas in part (a), or multiply the length of the rectangle by the width 2๐‘ง(๐‘ง 8) 2๐‘ง 2 16๐‘ง. Both give the same final result.The distributive propertyWhat would be another way to find the total area? Finding the area of the two separate rectangles and adding their areas: 2๐‘ง(๐‘ง) 2๐‘ง(8) 2๐‘ง 2 16๐‘งScaffolding:Model the following examples of polynomial multiplication for students who need to review.Multiply two monomials: Point out to students that because multiplication is both commutative and associative, factors maybe reordered to group the numerical factors together and then the variable factors together.5๐‘Ž๐‘ 4๐‘ (5 4)(๐‘Ž๐‘ ๐‘) 20๐‘Ž๐‘๐‘3๐‘ฅ 2 4๐‘ฅ 3 ๐‘ฆ 5๐‘ฆ 2 (3 4 5)(๐‘ฅ 2 ๐‘ฅ 3 ๐‘ฆ1 ๐‘ฆ 2 ) 60๐‘ฅ 2 3 ๐‘ฆ1 2 60๐‘ฅ 5 ๐‘ฆ 3Multiply a polynomial by a monomial: Some students may benefit from relating multiplication of polynomials tomultiplication of numbers in base 10. In the example below, the multiplication process is represented vertically (like a base10 product of a 2-digit number by a 1-digit number) and then horizontally, using the distributive property.Multiply (5๐‘Ž 7๐‘) by 3๐‘. To find this product vertically, follow the same procedure as you would with place values forwhole numbers. Just be sure to follow the rules for combining like terms. Show how to multiply vertically. 5๐‘Ž 7๐‘ 3๐‘15๐‘Ž๐‘ 21๐‘๐‘Now, multiply the polynomial by the monomial horizontally, using the distributive property for multiplication over addition.Make sure each term of the first binomial is distributed over both terms of the second.(5๐‘Ž 7๐‘)3๐‘ (5๐‘Ž 3๐‘) (7๐‘ 3๐‘) 15๐‘Ž๐‘ 21๐‘๐‘(The associative property for multiplication allows us to group the numbers and the variables together.)Lesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201518This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Lesson 1NYS COMMON CORE MATHEMATICS CURRICULUMM4ALGEBRA IExample 1 (3 minutes)Have students work on this example with a partner or in small groups. Have the groups share their processes and theirfindings and discuss the differences in processes used (if there are any).Example 1Jackson has given his friend a challenge:The area of a rectangle, in square units, is represented by ๐Ÿ‘๐’‚๐Ÿ ๐Ÿ‘๐’‚ for some real number ๐’‚. Find the length and width ofthe rectangle.How many possible answers are there for Jacksonโ€™s challenge to his friend? List the answer(s) you find.๐Ÿ‘๐’‚๐Ÿ ๐Ÿ‘๐’‚ square units๐Ÿ‘๐’‚๐Ÿ ๐Ÿ‘๐’‚ ๐Ÿ‘๐’‚(๐’‚ ๐Ÿ) so the width of the rectangle could be ๐Ÿ‘๐’‚ units and the length could be (๐’‚ ๐Ÿ) units.๐Ÿ๐Ÿ๐Ÿ๐Ÿ(Students may opt to factor only ๐’‚ or ๐Ÿ‘ or even ๐’‚: ๐’‚(๐Ÿ‘๐’‚ ๐Ÿ‘) or ๐Ÿ‘(๐’‚๐Ÿ ๐’‚) or๐’‚(๐Ÿ”๐’‚ ๐Ÿ”).)There are infinite representations for the dimensions of the rectangle.If students try to use 1 as the common factor for two or more numbers, point out that, while 1 is indeed a factor,factoring out a 1 does not help in finding the factors of an expression. If this issue arises, it may be necessary to discussthe results when factoring out a 1.Factoring out the Greatest Common Factor (GCF)Students now revisit factoring out the greatest common factor as was introduced in Grade 6, Module 2. When factoring a polynomial, we first look for a monomial that is the greatest common factor (GCF) of all theterms of the polynomial. Then, we reverse the distribution process by factoring the GCF out of each term andwriting it on the outside of the parentheses. In the example above, we factored out the GCF: 3๐‘Ž.Exercises 1โ€“3 (3 minutes)For the exercises below, have students work with a partner or in small groups to factor out the GCF for each expression.Exercises 1โ€“3Factor each by factoring out the greatest common factor:1.๐Ÿ๐ŸŽ๐’‚๐’ƒ ๐Ÿ“๐’‚๐Ÿ“๐’‚(๐Ÿ๐’ƒ ๐Ÿ)Lesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201519This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Lesson 1NYS COMMON CORE MATHEMATICS CURRICULUMM4ALGEBRA I2.๐Ÿ‘๐’ˆ๐Ÿ‘ ๐’‰ ๐Ÿ—๐’ˆ๐Ÿ ๐’‰ ๐Ÿ๐Ÿ๐’‰๐Ÿ‘๐’‰(๐’ˆ๐Ÿ‘ ๐Ÿ‘๐’ˆ๐Ÿ ๐Ÿ’)3.๐Ÿ”๐’™๐Ÿ ๐’š๐Ÿ‘ ๐Ÿ—๐’™๐’š๐Ÿ’ ๐Ÿ๐Ÿ–๐’š๐Ÿ“๐Ÿ‘๐’š๐Ÿ‘ (๐Ÿ๐’™๐Ÿ ๐Ÿ‘๐’™๐’š ๐Ÿ”๐’š๐Ÿ ) (Students may find this one to be more difficult. It is used as an example in a scaffoldingsuggestion below.)Discussion (4 minutes): The Language of PolynomialsMake sure students have a clear understanding of the following terms and use them appropriately during instruction.The scaffolding suggestion below may be used to help students understand the process of factoring out the GCF. Beginthe discussion by reviewing the definition of prime and composite numbers given in the student materials.Discussion: The Language of PolynomialsPRIME NUMBER: A prime number is a positive integer greater than ๐Ÿ whose only positive integer factors are ๐Ÿ and itself.COMPOSITE NUMBER: A composite number is a positive integer greater than ๐Ÿ that is not a prime number.A composite number can be written as the product of positive integers with at least one factor that is not ๐Ÿ or itself.For example, the prime number ๐Ÿ• has only ๐Ÿ and ๐Ÿ• as its factors. The composite number ๐Ÿ” has factors of ๐Ÿ, ๐Ÿ, ๐Ÿ‘, and ๐Ÿ”;it could be written as the product ๐Ÿ ๐Ÿ‘. Factoring is the reverse process of multiplication (through multiple use of the distributive property). We factora polynomial by reversing the distribution processโ€”factoring the GCF out of each term and writing it on theoutside of the parentheses. To check whether the polynomial's factored form is equivalent to its expandedform, you can multiply the factors to see if the product yields the original polynomial. 4(๐‘ฅ 3) is called a factored form of 4๐‘ฅ 12.A nonzero polynomial expression with integer coefficients is said to be prime or irreducible over the integers if it satisfiestwo conditions:(1)It is not equivalent to ๐Ÿ or ๐Ÿ, and(2)If the polynomial is written as a product of two polynomial factors, each with integer coefficients, then one ofthe two factors must be ๐Ÿ or ๐Ÿ. Note that this definition actually specifies prime numbers and their negatives as well (the case when thepolynomial has degree 0). For example: 4๐‘ฅ 9 is irreducible over the integers.Given a polynomial in standard form with integer coefficients, let ๐’„ be the greatest common factor of all of thecoefficients. The polynomial is factored completely over the integers when it is written as a product of ๐’„ and one or moreprime polynomial factors, each with integer coefficients. In the future, we learn to factor over the rationals and reals.Lesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201520This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

NYS COMMON CORE MATHEMATICS CURRICULUMLesson 1M4ALGEBRA IScaffolding:For students who struggle with factoring the GCF from a more complicated polynomial, suggest they use a chart to organizethe terms and factors. Here is an example using Exercise 3 above:Stack the three terms (monomial expressions) on the far left of a table, and then write each of the terms of the polynomial inprime factor form across the row, stacking those that are the same. Then, shade or circle the columns that have the samefactor for all three terms.Now, look down the columns to find which factors are in all three rows. The blue columns show those common factors,which are shared by all three terms. So, the greatest common factor (GCF) for the three terms is the product of thosecommon factors: 3๐‘ฆ 3 . This term is written on the outside of the parentheses. Then reversing the distributive property, wewrite the remaining factors inside the parentheses for each of the terms that are not in the blue shaded columns. (It may behelpful to point out that factoring out the GCF is the same as dividing each term by the GCF.) In this example, it is3๐‘ฆ 3 (2๐‘ฅ 2 3๐‘ฅ๐‘ฆ 6๐‘ฆ 2 ). You can find the GCF by multiplying the factors across the bottom, and you can find the terms ofthe other factor by multiplying the remaining factors across each row.Example 2 (4 minutes): Multiply Two BinomialsDemonstrate that the product can be found by applying the distributive property (twice) where the first binomialdistributes over each of the second binomialโ€™s terms, and relate the result to the area model as was used in Module 1and as is shown below. Note that while the order of the partial products shown corresponds with the well-known FOILmethod (Firsts, Outers, Inners, Lasts), teachers are discouraged from teaching polynomial multiplication as a procedureor with mnemonic devices such as FOIL. Instead, foster understanding by relating the process to the distributiveproperty and the area model.Since side lengths of rectangles cannot be negative, it is not directly applicable to use the area model for multiplyinggeneral polynomials. (We cannot be certain that each term of the polynomial represents a positive quantity.) However,we can use a tabular method that resembles the area model to track each partial product as we use the distributiveproperty to multiply the polynomials.Example 2: Multiply Two BinomialsUsing a Table as an AidYou have seen the geometric area model used in previous examples to demonstrate the multiplication of polynomialexpressions for which each expression was known to represent a measurement of length.Without a context such as length, we cannot be certain that a polynomial expression represents a positive quantity.Lesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201521This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Lesson 1NYS COMMON CORE MATHEMATICS CURRICULUMM4ALGEBRA ITherefore, an area model is not directly applicable to all polynomial multiplication problems. However, a table can beused in a similar fashion to identify each partial product as we multiply polynomial expressions. The table serves toremind us of the area model even though it does not represent area.For example, fill in the table to identify the partial products of (๐’™ ๐Ÿ)(๐’™ ๐Ÿ“). Then, write the product of (๐’™ ๐Ÿ)(๐’™ ๐Ÿ“)in standard form.๐’™๐’™ ๐Ÿ ๐Ÿ“๐’™๐Ÿ๐Ÿ“๐’™๐Ÿ๐’™๐Ÿ๐ŸŽ๐’™๐Ÿ ๐Ÿ•๐’™ ๐Ÿ๐ŸŽWithout the Aid of a TableRegardless of whether or not we make use of a table as an aid, the multiplying of two binomials is an application of thedistributive property. Both terms of the first binomial distribute over the second binomial. Try it with (๐’™ ๐’š)(๐’™ ๐Ÿ“). Inthe example below, the colored arrows match each step of the distribution with the resulting partial product.๐’™๐Ÿ ๐Ÿ“๐’™๐Œ๐ฎ๐ฅ๐ญ๐ข๐ฉ๐ฅ๐ฒ: (๐’™ ๐’š)(๐’™ ๐Ÿ“) ๐’š๐’™๐’™๐Ÿ ๐Ÿ“๐’™ ๐’š๐’™ ๐Ÿ“๐’š ๐Ÿ“๐’š}Example 3 (4 minutes): The Difference of SquaresExample 3: The Difference of SquaresFind the product of (๐’™ ๐Ÿ)(๐’™ ๐Ÿ). Use the distributive property to distribute the first binomial over the second.With the Use of a Table:๐’™๐’™ ๐Ÿ ๐Ÿ๐’™๐Ÿ๐Ÿ๐’™ ๐Ÿ๐’™ ๐Ÿ’๐’™๐Ÿ ๐Ÿ’Without the Use of a Table:(๐’™)(๐’™) (๐’™)( ๐Ÿ) (๐Ÿ)(๐’™) (๐Ÿ)( ๐Ÿ) ๐’™๐Ÿ ๐Ÿ๐’™ ๐Ÿ๐’™ ๐Ÿ’ ๐’™๐Ÿ ๐Ÿ’Lesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201522This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

NYS COMMON CORE MATHEMATICS CURRICULUMLesson 1M4ALGEBRA I Do you think the linear terms are always opposites when we multiply the sum and difference of the same twoterms? Why? Yes. When we multiply the first term of the first binomial by the last term of the second, we get theopposite of what we get when we multiply the second term of the first binomial by the first term of thesecond.So, ๐‘ฅ 2 4 is the difference of two perfect squares. Factoring the difference of two perfect squares reversesthe process of finding the product of the sum and difference of two terms.Exercise 4 (6 minutes)The following can be used as a guided practice or as independent practice.Exercise 4Factor the following examples of the difference of perfect squares.a.b.๐’•๐Ÿ ๐Ÿ๐Ÿ“(๐’• ๐Ÿ“)(๐’• ๐Ÿ“)๐Ÿ๐Ÿ’๐’™ ๐Ÿ—๐Ÿ(๐Ÿ๐’™ ๐Ÿ‘)(๐Ÿ๐’™ ๐Ÿ‘)๐Ÿc.๐Ÿ๐Ÿ”๐’‰ ๐Ÿ‘๐Ÿ”๐’Œd.๐Ÿ๐Ÿ’ ๐’ƒ๐Ÿ’(๐Ÿ’๐’‰ ๐Ÿ”๐’Œ)(๐Ÿ’๐’‰ ๐Ÿ”๐’Œ)(๐Ÿ ๐’ƒ)(๐Ÿ ๐’ƒ)e.๐’™ ๐Ÿ’(๐’™๐Ÿ ๐Ÿ)(๐’™๐Ÿ ๐Ÿ)f.๐’™๐Ÿ” ๐Ÿ๐Ÿ“(๐’™๐Ÿ‘ ๐Ÿ“)(๐’™๐Ÿ‘ ๐Ÿ“)Point out that any even power can be a perfect square and that 1 is always a square.Write a General Rule for Finding the Difference of Squares๐Ÿ๐ŸWrite ๐’‚ ๐’ƒ in factored form.(๐’‚ ๐’ƒ)(๐’‚ ๐’ƒ)Exercises 5โ€“7 (4 minutes)The following exercises may be guided or modeled, depending on how well students did on the previous example.Exercises 5โ€“7Factor each of the following differences of squares completely:5.๐Ÿ—๐’š๐Ÿ ๐Ÿ๐ŸŽ๐ŸŽ๐’›๐Ÿ(๐Ÿ‘๐’š ๐Ÿ๐ŸŽ๐’›)(๐Ÿ‘๐’š ๐Ÿ๐ŸŽ๐’›)6.๐’‚๐Ÿ’ ๐’ƒ๐Ÿ”(๐’‚๐Ÿ ๐’ƒ๐Ÿ‘ )(๐’‚๐Ÿ ๐’ƒ๐Ÿ‘ )7.๐’“๐Ÿ’ ๐Ÿ๐Ÿ”๐’”๐Ÿ’ (Hint: This one factors twice.)(๐’“๐Ÿ ๐Ÿ’๐’”๐Ÿ )(๐’“๐Ÿ ๐Ÿ’๐’”๐Ÿ ) (๐’“๐Ÿ ๐Ÿ’๐’”๐Ÿ )(๐’“ ๐Ÿ๐’”)(๐’“ ๐Ÿ๐’”)Lesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201523This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Lesson 1NYS COMMON CORE MATHEMATICS CURRICULUMM4ALGEBRA IExample 4 (4 minutes): The Square of a BinomialIt may be worthwhile to let students try their hands at finding the product before showing them how. If studentsstruggle to include every step in the process, pause at each step so that they have time to absorb the operations thattook place.Example 4: The Square of a BinomialTo square a binomial, such as (๐’™ ๐Ÿ‘)๐Ÿ , multiply the binomial by itself.(๐’™ ๐Ÿ‘)(๐’™ ๐Ÿ‘) (๐’™)(๐’™) (๐’™)(๐Ÿ‘) (๐Ÿ‘)(๐’™) (๐Ÿ‘)(๐Ÿ‘) ๐’™๐Ÿ ๐Ÿ‘๐’™ ๐Ÿ‘๐’™ ๐Ÿ— ๐’™๐Ÿ ๐Ÿ”๐’™ ๐Ÿ—Square the following general examples to determine the general rule for squaring a binomial:a.(๐’‚ ๐’ƒ)๐Ÿ(๐’‚ ๐’ƒ)(๐’‚ ๐’ƒ) ๐’‚๐Ÿ ๐’‚๐’ƒ ๐’ƒ๐’‚ ๐’ƒ๐Ÿ ๐’‚๐Ÿ ๐Ÿ๐’‚๐’ƒ ๐’ƒ๐Ÿb.(๐’‚ ๐’ƒ)๐Ÿ(๐’‚ ๐’ƒ)(๐’‚ ๐’ƒ) ๐’‚๐Ÿ ๐’‚๐’ƒ ๐’ƒ๐’‚ ๐’ƒ๐Ÿ ๐’‚๐Ÿ ๐Ÿ๐’‚๐’ƒ ๐’ƒ๐ŸPoint out that the process used in squaring the binomial is called expanding; in general, expanding means rewriting aproduct of sums as a sum of products through use of the distributive property. How are the answers to the two general examples similar? How are they different? What is the cause of thedifference between the two? Both results are quadratic expressions with three terms. The first and second examples both have an๐‘Ž2 , ๐‘ 2 , and a 2๐‘Ž๐‘ term. In part b, the 2๐‘Ž๐‘ term is negative, while it is positive in part a. The negative(subtraction) in part b causes the middle term to be negative.Exercises 8โ€“9 (3 minutes)Exercises 8โ€“9Square the binomial.8.(๐’‚ ๐Ÿ”)๐Ÿ๐’‚๐Ÿ ๐Ÿ๐Ÿ๐’‚ ๐Ÿ‘๐Ÿ”9.(๐Ÿ“ ๐’˜)๐Ÿ๐Ÿ๐Ÿ“ ๐Ÿ๐ŸŽ๐’˜ ๐’˜๐ŸLesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201524This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Lesson 1NYS COMMON CORE MATHEMATICS CURRICULUMM4ALGEBRA IClosing (2 minutes) Factoring is the reverse process of multiplication. Look for a GCF first when you are factoring a polynomial. Keep factoring until all the factors are prime. Factor the difference of squares ๐‘Ž2 ๐‘ 2 as (๐‘Ž ๐‘)(๐‘Ž ๐‘).Lesson SummaryFactoring is the reverse process of multiplication. When factoring, it is always helpful to look for a GCF that can bepulled out of the polynomial expression. For example, ๐Ÿ‘๐’‚๐’ƒ ๐Ÿ”๐’‚ can be factored as ๐Ÿ‘๐’‚(๐’ƒ ๐Ÿ).Factor the difference of perfect squares ๐’‚๐Ÿ ๐’ƒ๐Ÿ :(๐’‚ ๐’ƒ)(๐’‚ ๐’ƒ).When squaring a binomial (๐’‚ ๐’ƒ),(๐’‚ ๐’ƒ)๐Ÿ ๐’‚๐Ÿ ๐Ÿ๐’‚๐’ƒ ๐’ƒ๐Ÿ .When squaring a binomial (๐’‚ ๐’ƒ),(๐’‚ ๐’ƒ)๐Ÿ ๐’‚๐Ÿ ๐Ÿ๐’‚๐’ƒ ๐’ƒ๐Ÿ .Exit Ticket (4 minutes)Lesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201525This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Lesson 1NYS COMMON CORE MATHEMATICS CURRICULUMM4ALGEBRA INameDateLesson 1: Multiplying and Factoring Polynomial ExpressionsExit TicketWhen you multiply two terms by two terms, you should get four terms. Why is the final result when you multiply twobinomials sometimes only three terms? Give an example of how your final result can end up with only two terms.Lesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201526This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

NYS COMMON CORE MATHEMATICS CURRICULUMLesson 1M4ALGEBRA IExit Ticket Sample SolutionsWhen you multiply two terms by two terms, you should get four terms. Why is the final result when you multiply twobinomials sometimes only three terms? Give an example of how your final result can end up with only two terms.Often when you multiply two binomials, each has a term with the same variable, say ๐’™, and two of the terms combine tomake one single ๐’™-term. If the two terms combine to make zero, there will be only two of the four terms left. Forexample, (๐’™ ๐Ÿ‘)(๐’™ ๐Ÿ‘) ๐’™๐Ÿ ๐Ÿ—.Problem Set Sample Solutions1.For each of the following, factor out the greatest common factor:a.๐Ÿ”๐’š๐Ÿ ๐Ÿ๐Ÿ–๐Ÿ”(๐’š๐Ÿ ๐Ÿ‘)b.๐Ÿ๐Ÿ•๐’š๐Ÿ ๐Ÿ๐Ÿ–๐’š๐Ÿ—๐’š(๐Ÿ‘๐’š ๐Ÿ)c.๐Ÿ๐Ÿ๐’ƒ ๐Ÿ๐Ÿ“๐’‚๐Ÿ‘(๐Ÿ•๐’ƒ ๐Ÿ“๐’‚)d.๐Ÿ๐Ÿ’๐’„๐Ÿ ๐Ÿ๐’„๐Ÿ๐’„(๐Ÿ•๐’„ ๐Ÿ)e.๐Ÿ‘๐’™๐Ÿ ๐Ÿ๐Ÿ•๐Ÿ‘(๐’™๐Ÿ ๐Ÿ—)2.Multiply.a.(๐’ ๐Ÿ“)(๐’ ๐Ÿ“)๐’๐Ÿ ๐Ÿ๐Ÿ“b.(๐Ÿ’ ๐’š)(๐Ÿ’ ๐’š)๐Ÿ๐Ÿ” ๐’š๐Ÿc.(๐’Œ ๐Ÿ๐ŸŽ)๐Ÿ๐’Œ๐Ÿ ๐Ÿ๐ŸŽ๐’Œ ๐Ÿ๐ŸŽ๐ŸŽd.(๐Ÿ’ ๐’ƒ)๐Ÿ๐Ÿ๐Ÿ” ๐Ÿ–๐’ƒ ๐’ƒ๐ŸLesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201527This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Lesson 1NYS COMMON CORE MATHEMATICS CURRICULUMM4ALGEBRA I3.The measure of a side of a square is ๐’™ units. A new square is formed with each side ๐Ÿ” units longer than the originalsquareโ€™s side. Write an expression to represent the area of the new square. (Hint: Draw the new square and countthe squares and rectangles.)Original square๐’™New square:(๐’™ ๐Ÿ”)๐Ÿ or ๐’™๐Ÿ ๐Ÿ๐Ÿ๐’™ ๐Ÿ‘๐Ÿ”4.In the accompanying diagram, the width of the inner rectangle is represented by ๐’™ ๐Ÿ‘ and the length by ๐’™ ๐Ÿ‘. Thewidth of the outer rectangle is represented by ๐Ÿ‘๐’™ ๐Ÿ’ and the length by ๐Ÿ‘๐’™ ๐Ÿ’.๐’™ ๐Ÿ‘๐Ÿ‘๐’™ ๐Ÿ’๐’™ ๐Ÿ‘๐Ÿ‘๐’™ ๐Ÿ’a.Write an expression to represent the area of the larger rectangle.Find the area of the larger (outer) rectangle by multiplying the binomials:(๐Ÿ‘๐’™ ๐Ÿ’)(๐Ÿ‘๐’™ ๐Ÿ’) ๐Ÿ—๐’™๐Ÿ ๐Ÿ๐Ÿ”.b.Write an expression to represent the area of the smaller rectangle.Find the area of the smaller (inner) rectangle by multiplying the binomials:(๐’™ ๐Ÿ‘)(๐’™ ๐Ÿ‘) ๐’™๐Ÿ ๐Ÿ—.c.Express the area of the region inside the larger rectangle but outside the smaller rectangle as a polynomial interms of ๐’™. (Hint: You will have to add or subtract polynomials to get your final answer.)Subtract the area of the smaller rectangle from the area of the larger rectangle:(๐Ÿ—๐’™๐Ÿ ๐Ÿ๐Ÿ”) (๐’™๐Ÿ ๐Ÿ—) ๐Ÿ—๐’™๐Ÿ ๐Ÿ๐Ÿ” ๐’™๐Ÿ ๐Ÿ— ๐Ÿ–๐’™๐Ÿ ๐Ÿ•.Lesson 1:Multiplying and Factoring Polynomial ExpressionsThis work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. eureka-math.orgThis file derived from ALG I-M4-TE-1.3.0-09.201528This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

property to multiply the polynomials. Example 2: Multiply Two Binomials Using a Table as an Aid You have seen the geometric area model used in previous examples to demonstrate the multiplication of polynomial expressions for which each e

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