Topic 23: Polynomial addition and multiplication 377 POLYNOMIAL ADDITION AND MULTIPLICATION Lesson 23.1 Introduction to polynomials 23.1 OPENER The Bradley family has decided to build a house in a new subdivision in Smithville. The Bradleys have already purchased a 150- foot by 200- foot lot on which to construct the new home. They start by looking at a plan of the lot. 1. What is the area of the Bradleys lot? The subdivision developer was given guidelines for building in the area. Two guidelines apply to the placement of the house on the lot. First, the builder must leave a 20- foot easement on either side of the house to accommodate the placement of utilities. Second, the builder must locate the house at least 50 feet back from the front of the lot, to allow for sidewalks, streetlights, and landscaping. 2. What is the area of the portion of the lot that the Bradleys can use for building their house? 23.1 CORE ACTIVITY 1. Will a house with the dimensions shown here fit in the allowable space on the Bradleys lot? How do you know? 2. What does x represent in the Bradleys house plan?
378 Unit 8 Quadratic functions and equations 3. The building cost is $110 per square foot, and the Bradleys budget is $200,000. Complete the table to determine the area of the house and the cost to build it for different values of x. How large can x be for the house to still fit on the lot and be within the Bradleys budget? Value for x Length of house Width of house Total area of house Cost to build house Within budget? 0 5 10 4. What is a polynomial? 5. What are terms of a polynomial? 6. Complete the following table to describe different polynomials. Type Definition Example monomial 5mn 2 5xy 7 a polynomial with three terms 7. What is the degree of a monomial? 8. What is the degree of 3x 2 y 4? 9. What is the degree of a polynomial? 10. What is the degree of 7x 4 + 5x 3? 11. Using the answer choices provided, fill in the table with the correct attributes for each polynomial. 3 monomial 1 binomial 2 trinomial 4 Polynomial Degree Special name 3x 2 75 x 3 + 3x 2 + x 3x 2 yz 6a 2 + 2ab 2b 2 4x 3
Topic 23: Polynomial addition and multiplication 379 23.1 REVIEW ONLINE ASSESSMENT You will work with your class to review the online assessment questions. Problems we did well on Problems we need to review Addressing areas of incomplete understanding Use this page and notebook paper to take notes and re- work particular online assessment problems that your class identifies. Problem # Work for problem: Problem # Work for problem: Problem # Work for problem:
380 Unit 8 Quadratic functions and equations HOMEWORK 23.1 Notes or additional instructions based on whole- class discussion of homework assignment: Next class period, you will take a mid- unit assessment. One good study skill to prepare for tests is to review the important skills and ideas you have learned. Use this list to help you review these skills and concepts, especially by reviewing related course materials. Important skills and ideas from Topic 22: Simplify expressions containing radicals Approximate the value of a radical expression Use properties of quadratic functions to answer questions from data Solve quadratic equations by graphing Use tables to solve quadratic equations Solve quadratic equations using the quadratic formula Use the discriminant to determine the number of solutions of a quadratic equation Homework Assignment Part I: Study for the mid- unit assessment by reviewing the key ideas listed above. Part II: Complete the online More practice in the topic Quadratic models and equations. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study. Part III: Complete the Staying Sharp 23.1.
Topic 23: Polynomial addition and multiplication 381 STAYING SHARP 23.1 Reviewing pre- algebra ideas 1. Here are six natural numbers: 2, 6, 11, 15, 19, 24 a. Circle each prime number. Underline each composite number. b. Explain why 11 is prime: c. Explain why 15 is composite: 2. Consider this Square Box Problem. Explain the relationship between the two numbers in the left and right parts of the box and the number at the top. Explain the relationship between the left and right numbers and the number at the bottom. 3. Rewrite this expression as simply as you can: 4x + 5y + 2x + 6z + 3x + 11y + 7x + 9z 4. Rewrite this expression with only two exponents: (3 x 2 y 5 ) (7 x 4 y 6 ) Practicing algebra skills & concepts 5. Complete the parts of this area model to find the product of 3 23 = 3 (20 + 3): 20 3 6. Rewrite this expression without parentheses: 3(2x + 3) Preparing for upcoming lessons 3
382 Unit 8 Quadratic functions and equations
Topic 23: Polynomial addition and multiplication 383 Lesson 23.2 Checking for understanding 23.2 OPENER 2 Solve the quadratic equation x + 5x + 6= 0 using the method specified. 1. By graphing (using either your graphing calculator or graphing by hand). 2. By using the quadratic formula, ± x = 2 b b 4ac 2a. Solutions: 23.2 MID-UNIT ASSESSMENT 23.2 CONSOLIDATION ACTIVITY 1. Leyla and Yusuf s teacher gives them three polynomial expressions and asks them to tell what degree each polynomial expression is. For each polynomial expression, only one student gives the correct answer. Circle the correct answers and provide advice that might help either Leyla or Yusuf. Polynomial expression Degree according to Leyla Degree according to Yusuf Advice to help the student who answered incorrectly 9x 2 7x + 4 2 9 3a 2 b 5 c 7 8 x 5 + 4x 3 y 2 10 5 2. As you did in the last lesson, classify each polynomial using its special name. Polynomial Special name monomial binomial trinomial 9x 2 7x + 4 3a 2 b 5 c x 5 + 4x 3 y 2
384 Unit 8 Quadratic functions and equations 3. Here is a basic floor plan for Robin s apartment, with measurements provided in feet. Each expression in the following table describes the total area of two or more rooms in Robin s apartment. Place a check mark for the rooms whose total areas are described by the expression. The first row is completed for you. Expression Living Room Bedroom Kitchen Bathroom 10 15 + 8 12 ü ü 12 10 + 12 8 12 (10 + 8) 10 15 + 10 12 + 8 15 + 8 12 10(15 + 12) + 8(15 + 12) (10 + 8) (15 + 12) 4. Here is a basic floor plan for Marcos s apartment, with measurements provided in feet. Write an expression for each row in the table. Each expression should describe the total area of the rooms marked in that row. Expression Living Room Bedroom Kitchen Bathroom ü ü ü ü ü ü ü ü
Topic 23: Polynomial addition and multiplication 385 HOMEWORK 23.2 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Complete the following Math Journal: Term My understanding of this term An example that illustrates this term Monomial Binomial Trinomial Polynomial Degree of a polynomial 2. Write the expression that matches the collection of algebra tiles. 3. Draw algebra tiles to model the expression 2 3x 4x + 7.
386 Unit 8 Quadratic functions and equations 4. Here is a basic floor plan for Jaime s apartment with measurements in feet. a. Suppose x = 5 feet. For each row in the table, find the total area of the checkmarked rooms. Living room ü Kitchen ü Dining room ü ü ü ü Bedroom Bathroom Closet AREA ü ü ü ü ü ü ü ü ü b. Suppose x is unknown. For each row in the table, write an expression that describes the total area of the checkmarked rooms. Simplify each expression as much as possible. Living room Kitchen Dining room ü ü ü Bedroom Bathroom Closet AREA ü ü ü ü ü ü ü ü ü ü ü ü
Topic 23: Polynomial addition and multiplication 387 STAYING SHARP 23.2 1. Here is a factor tree for 48. 2. Complete this Square Box Problem: Reviewing pre- algebra ideas Make a factor tree for 120: 11 2 6 3. Identify the error in each statement. a. 2x + 2y = 4xy 4. Rewrite this expression with a single exponent: (3 x 4 ) 2 Practicing algebra skills & concepts b. 3x + 4 = 7x c. 4x 2 + 5x = 9x 3 5. Label this area model to represent 3 23 = 3 (30 7). 6. Rewrite this expression without parentheses: 4 (3x 5) Preparing for upcoming lessons 3
388 Unit 8 Quadratic functions and equations
Topic 23: Polynomial addition and multiplication 389 Lesson 23.3 Polynomial products 23.3 OPENER The laundry room in the Bradleys' modified house plan is x + 8 feet by x + 7 feet. Find the area of the laundry room for each of the following values of x: 1. 5 feet 2. 10 feet 3. 20 feet 23.3 CORE ACTIVITY 1. Complete the following figure to show how algebra tiles can be used to model (x + 8) (x + 7). Write an answer for this problem.
390 Unit 8 Quadratic functions and equations 2. Use an area model to multiply (5 + 8) (5 + 7). 3. Use an area model to multiply (10 + 8) (10 + 7). 4. Use an area model to multiply (20 + 8) (20 + 7). 5. Use an area model to multiply (x + 8) (x + 7). Simplify as much as possible. 6. Multiply (x + 8) (x + 7) by directly applying the distributive property. 7. Use an area model to find the volume of a rectangular prism with a base of area B = (x 2 + 5x + 4) in 2 and a height of h = (x + 3) in. 23.3 CONSOLIDATION ACTIVITY 1. Here is a basic floor plan for Andrea s apartment with measurements provided in feet. 2x 9 a. Write an expression for the area of each room. Be sure that the expression is simplified. x Living room Bedroom Living room Bedroom 7 Kitchen Bathroom Kitchen Bathroom b. Write an expression for the area of Andrea s entire apartment by adding the room areas. Combine like terms.
Topic 23: Polynomial addition and multiplication 391 c. Show how to directly apply the distributive property to multiply the two binomials. (2x + 9)(x + 7) d. If x = 6, what is the area of Andrea s entire apartment? Explain how you found your answer. 2. In the previous activity, you used an area model to multiply a trinomial and a binomial to find the volume of a prism. Complete the given area models to multiply the following polynomials with different numbers of terms. Binomial times a binomial: Monomial times a binomial: Trinomial times a binomial: a. ( x+ 3) ( x+ 7) = b. x ( x 8) = c. ( x+ y+ z) ( x+ 2) = 3. The formula for finding the volume of a rectangular prism is V = Bh, where B is the base area and h is the height. Consider a rectangular prism that has a square base with side length of x + 5 and a height of x + 2. a. Use an area model to help you find an expression for the base area, B. Combine like terms. B = b. Now, use an area model to help you find an expression for the volume of the prism. V =
392 Unit 8 Quadratic functions and equations 4. The dimensions of the largest room in the Bradleys house are 2x + 17feet by 3x + 23feet. As shown on the floor plan, multiplying these dimensions gives an area of 6x 2 + 97x + 391 square feet. Find the areas for the remaining rooms. Show your work and write your answers on the floor plan.
Topic 23: Polynomial addition and multiplication 393 HOMEWORK 23.3 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Use area models to multiply the following polynomials. For the first three products, blank area models are provided. a. ( x+ 1) ( x + 3) b. ( x+ 5) ( x 4) c. x ( x + 11) d. ( x 10) ( x 6) e. (2x+ 2) (3x + 3) f. ( a+ b+ c) ( x + 5) 2. Juliette multiplied two polynomials. Her work is shown here: a. Did Juliette multiply correctly? If not, explain to Juliette what she did wrong and show how she can correct her work. ( x 3) ( x + 4) ( x) ( x) + ( 3) ( 4) 2 x 12 b. Show how an area model could have helped Juliette organize her work. 3. Here is the floor plan of an apartment. Find the area of each room. Room Area Living room Bathroom Closet Kitchen Bedroom
394 Unit 8 Quadratic functions and equations 4. Multiply the following polynomials. You can use any method. a. 3 x 2 g ( x 2 4x+ 8) b. (2x 7) g(2x + 5) c. ( x 2 1) g ( x 2 + 2x 3) 5. Find the volume of a rectangular prism with a base of area B= x 2 + 5x+ 4 in 2 and a height of h= x+ 3 in.
Topic 23: Polynomial addition and multiplication 395 STAYING SHARP 23.3 1. Find all of the factors of 60. 2. Complete this Square Box Problem: Reviewing pre- algebra ideas 24 3. Line m passes through points ( 3,5) and (1, 2). What is the slope of line m? 4. a. Rewrite this expression 11 without parentheses: 3(5x + 2y) Practicing algebra skills & concepts b. Rewrite this expression without parentheses: (x 5 y 2 ) 3 c. What do you notice about your two answers? Preparing for upcoming lessons 5. Complete this area model to find the product of 14 16. 10 4 10 6 6. Fill in the missing parts of this area model of multiplication. 10 3 10 100 15
396 Unit 8 Quadratic functions and equations
Topic 23: Polynomial addition and multiplication 397 Lesson 23.4 Making connections 23.4 OPENER Area models and algebra tiles are two models that can help you multiply polynomials. Both models are shown for the multiplication problem ( x+ 3)( x + 2). Note that grid lines have been added to the algebra tile workspace to help you see the connection between the models. Note also that the multiplication dot does not appear in this multiplication problem. In other words: ( x+ 3) ( x + 2) = ( x+ 3)( x + 2). Area Model Algebra Tiles Describe ways in which the two models are similar. 23.4 CORE ACTIVITY 1. Complete the following figure to show how algebra tiles can be used to model (3x 2)(2x + 2). Write an answer for this problem. 2. Use an area model to simplify (3x 2)(2x + 2).
398 Unit 8 Quadratic functions and equations 3. Solve the same polynomial multiplication problem using three different methods, then compare the methods. a. Use algebra tiles to model and simplify (2x 3)(x + 1). (Draw tiles in the right- hand workspace.) b. Multiply (2x 3)(x + 1) using an area model. c. Multiply (2x 3)(x + 1) by directly applying the distributive property. d. Compare the processes and results of the algebra tile method, the area model method, and the method of directly applying the distributive property. 4. Use a method of your choice to find the area of the Bradleys family room. The dimensions of the room are 3x + 23 feet by 2x + 17 feet. 23.4 REVIEW MID-UNIT ASSESSMENT
Topic 23: Polynomial addition and multiplication 399 HOMEWORK 23.4 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Consider the algebra multiplication problem ( x+ 1)( x + 3). a. Show how algebra tiles can be used to find the product. Sketch the appropriate algebra tiles in the workspace. Then use your algebra tile model to write the product. ( x+ 1)( x+ 3) = b. Show how an area model can be used to find the product. c. Show how the product can be found by applying the distributive property directly. ( x+ 1)( x+ 3) = ( x+ 1)( x+ 3) = 2. Consider the algebra multiplication problem ( x+ 12 )( x 3). a. Show how algebra tiles can be used to find the product. Sketch the appropriate algebra tiles in the work space. Then use your algebra tile model to write the product. ( x+ 1)(2x 3) =
400 Unit 8 Quadratic functions and equations b. Show how an area model can be used to find the product. c. Show how the product can be found by applying the distributive property directly. ( x+ 1)(2x 3) = ( x+ 1)(2x 3) = 3. Compare the processes and results for multiplying polynomials using the algebra tile method, the area model method, and the method of directly applying the distributive property. How are these methods the same? How are they different? 4. Multiply the following polynomials using the method of your choice. a. ( x+ 7)( x + 9) b. ( x 5)( x 5) c. a ( a + 3) d. (5x+ 1) (6x + 3) e. (8x 2)(4 x + 3) f. ( x+ y+ 5)( y 2)
Topic 23: Polynomial addition and multiplication 401 STAYING SHARP 23.4 Reviewing pre- algebra ideas 1. List all of the factor pairs of 96. (For example, 1 and 96 are a pair, because 1 96 = 96.) 2. Complete the table of factor pairs of 120. Which pair of factors has a sum of 34? 1 120 2 4 6 40 24 12 Practicing algebra skills & concepts 3. Line k passes through points (3,2) and (11,4). Martin 11 3 8 calculates the slope as = = 4. 4 2 2 a. Identify the error in Martin s calculation. b. Find the correct slope of line k. Show your work. 4. a. Solve for x: 3(2x + 5) = 39 b. Check your solution in the original equation. 5. Complete this area model to find the product of 23 34. 30 4 6. Find the missing parts of this area model for multiplication. (Hint: Work backwards.) 7 Preparing for upcoming lessons 20 3 10 100 28
402 Unit 8 Quadratic functions and equations
Topic 23: Polynomial addition and multiplication 403 Lesson 23.5 Adding and subtracting polynomials 23.5 OPENER The guidelines for building on the Bradleys lot specify that there must be a 20- foot easement on each side of the house. The house must also be set back 50 feet. The Bradleys contractor, BK Builders, wants to know the perimeter of the entire lot, as well as the perimeter of the portion on which the home can be built. Using the diagram and what you know about perimeter, calculate both of these perimeters. 1. What is the perimeter of the Bradleys lot? 2. What is the perimeter of the portion of the lot that the Bradleys can use for building their house? 23.5 CORE ACTIVITY 1. Use the floor plan to find the perimeter of each of the following rooms. a. Room B b. Room I 2. What are like terms? 3. Use algebra tiles to simplify (2x 2 + 3x 1) + (4 6x).
404 Unit 8 Quadratic functions and equations 4. Use algebra tiles to simplify (3x 2 4x + 2) + (x 2 + 3x 4). Draw tiles in the workspaces shown here. 5. Simplify (7x 2 14x+ 25) + (3 x 2 + 3 x 4) 6. Simplify (3x 2 x + 2) + (16 x 2 ). 7. Find the perimeter of the Bradleys new house. 8. Simplify (3x 2 + 7x + 9) (7x 8). You can use the algebra tile model shown here to help you find the answer.
Topic 23: Polynomial addition and multiplication 405 9. Use algebra tiles to simplify (3x 2 + 2x 1) (x 2 4x 7). Remember to use the fact that subtraction is the inverse of addition. Draw tiles in the workspaces shown here. 10. Simplify (6x 2 7x+ 5) (3 x 2 + 4 x 1) 11. Simplify (5x 2 x + 2) (4 x 2 + x + 9). 12. A rectangular cooler holds x 3 + 11x 2 + 32x + 28 cubic inches of water. Matt accidentally leaves the plug out of the bottom of the cooler. By the time he replaces the plug, the volume of the water has decreased to x 3 x 2 + 12x 11 cubic inches. How much water was lost?
406 Unit 8 Quadratic functions and equations 23.5 CONSOLIDATION ACTIVITY Puzzle 1 To solve this puzzle, the sum of the algebraic expressions must be the same for all sides of the polygon. Complete the puzzle by writing an algebraic expression for each of the empty circles. Puzzle 2 To solve this puzzle, the sum of the algebraic expressions must be equal to 5x + 5 for all sides of the polygon. Complete the puzzle using the algebraic expressions provided. Use each expression only once.
Topic 23: Polynomial addition and multiplication 407 HOMEWORK 23.5 Notes or additional instructions based on whole- class discussion of homework assignment: Homework Assignment Part I: Complete the online More practice in the topic Polynomial addition and multiplication. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study. Part II: Complete the Staying Sharp 23.5.
408 Unit 8 Quadratic functions and equations STAYING SHARP 23.5 Reviewing pre- algebra ideas 1. Sometimes we are also interested in negative factors. List all of the factors of 68. (For example, 1 and 68 are factors because 1 68 = 68. 1 and 68 are also factors because 1 68 = 68) 2. Find a factor pair of 80 that has a sum of 21. 3. Find the solution of this system of two linear equations using substitution. 4. Find the solution of this system of two linear equations using linear combination. Practicing algebra skills & concepts 11x + 3y = 103 y = 3x + 1 2x + 7y = 77 5x + 7y = 115 5. Complete this area model to find the product (x + 3) (x + 4). 6. Find the missing parts of this area model for multiplication. (Hint: Work backwards.) Preparing for upcoming lessons x 3 x 4 4x x x 2 7x 4x 28