18.3 Special Products of Binomials PDF

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LESSON 18.3 18.3 Special Products of Binomials Special Products Essential Question: How can you find special products of binomials? of Binomials Resource Locker Common Core Math Standards Explore Modeling Special Products B. × The student is expected to: Use algebra tiles to model the special products of binomials. COMMON CORE A-APR.A.1 Use algebra tiles to model (2x + 3). Then write the product in simplest form. 2 Understand that polynomials form a system × × analogous to the integers, namely, they are closed under the operations of … multiplication… Also A-SSE.A.1, A-SSE.A.2 ? Mathematical Practices C. × COMMON CORE MP.7 Using Structure Language Objective Explain to a partner what a perfect square trinomial is. (2x + 3) = 2 ? x + 2 ? x+ ?. 4 12 9 Online Resources  Use algebra tiles to model (2x - 3). Then write  Use algebra tiles to model (2x + 3)(2x - 3). 2 An extra example for each Explain section is the product in simplest form. See tiles above. Then recount the tiles in the grid and write the expression. See tiles above. available online. © Houghton Mifflin Harcourt Publishing Company × × Engage ? ? Essential Question: How can you find special products of binomials? Use the special products rules for squaring a binomial (2x - 3) = 2 ? x - 2 ? x+ ?. (2x + 3)(2x - 3) = ? x + 2 ? x- ?. or for finding the product of a sum and a difference. 4 12 9 4 0 9 Preview: Lesson Performance Task Module 18 677 Lesson 3 View the Engage section online. Discuss what safety precautions one would need to keep in Professional Development mind when designing a fireplace. Then preview the Lesson Performance Task. Math Background Special products of binomials are predictable patterns that can be applied as rules to save time and effort when multiplying binomials. They are also useful in a variety of situations that students will encounter as they progress in math. For example, the binomials (a + b) and (a − b) are called conjugates. Their product is always a difference of squares, a2 − b2. One use of conjugates is to rewrite an expression to eliminate a square root in the denominator, as follows: ______ 5 + √3 ― 5 + √3 5 + √3 ― ― 1 = ______ 1  ______ = ________ = ______ 5 - √― 5 - √― ― 5 + √3 - ( √― 3) 22 2 3 3 5 2 Lesson 18.3 677  Reflect Explore 1. Discussion In Step A, which terms of the trinomial are perfect squares? What is the coefficient of x in the product? How can you use the values of a and b in the expression (2x + 3) to produce the coefficient of 2 each term in the trinomial? How can you generalize these results to write a rule for the product (a + b) ? 2 Modeling Special Products See below. 2. Discussion In Step B, which terms of the trinomial are perfect squares? What is the coefficient of x in the trinomial? How can you use the values of a and b in the expression (2x - 3) to produce the coefficient of 2 each term in the trinomial? How can you generalize these results to write a rule for the product (a - b) ? 2 See below. Integrate Technology 3. Discussion In Step C, which terms of the product are perfect squares? What is the coefficient of x in the product? How can you use the values of a and b in the expression (2x + 3)(2x - 3) to produce the Students have the option of completing the algebra coefficient of each term in the product? How can you generalize these results to write a rule for the product tiles activity either in the book or online. (a + b)(a- b)? 4x 2 and 9; 0; 4 = 2 2, 9 = 3 2, and 0 = 2(3) + 2(-3); (a - b)(a + b) = a 2 - b 2. Multiplying (a + b) 2 Explain 1 In the Explore, you determined a formula for the square of a binomial sum, (a + b) = a 2 + 2ab + b 2. A trinomial of 2 Integrate Mathematical Practices the form a 2 + 2ab + b 2 is called a perfect-square trinomial. A perfect-square trinomial is a trinomial that is the result Focus on Modeling of squaring a binomial. MP.4 Remind students that when modeling Example 1 Multiply. multiplication with algebra tiles, the product of a (x + 4) 2  (3x + 2y) 2 yellow (positive) tile and a red (negative) tile is a red tile, while the product of two yellow tiles or two red (a + b) 2 = a 2 + 2ab + b 2 (a + b) 2 = a 2 + 2ab + b 2 )( ) ( ) 2 tiles is a yellow tile. (x + 4) 2 = x 2 + 2(x)(4) + 4 2 (3x + 2y) = (3x) + 2 2 2 ( 3 x 2 y + 2 y = x 2 + 8x + 16 = 9x + 12 xy + 2 4 y 2 Reflect Explain 1 4. In the perfect square trinomial x 2 + bx + c, what is the relationship between b and c? Explain. (_b2 ) ; (a + b) = a + 2ab + b , and in this case, a = 1, 2ab = b, and b = c 2 2 Multiplying (a + b) 2 c= 2 2 2 2 © Houghton Mifflin Harcourt Publishing Company Your Turn Multiply. Questioning Strategies 5. (4 + x 2 ) 2 16 + 8x 2 + x 4 6. (-x + 3) 2 x 2 - 6x + 9 When squaring binomials, how many terms do you expect in the answer? Explain. Three terms; using the FOIL method, you will get four terms, but the 1. 4x 2 and 9; 12; 4 = 2 , 9 = 3 2, and 12 = 2(3) + 2(3) = 2(2)(3); (a + b) = a 2 + 2ab + b. 2 2 2 outer and inner products will be like terms that can 2. 4x 2 and 9; -12; 4 = 2 , 9 = 3 2, and -12 = 2(-3) + 2(-3) = -2(2)(3); (a - b) = a 2 - 2ab + b. 2 2 2 be combined, so the simplified result will have three terms. Why is the middle term of (a + b) equal to 2ab? If 2 you use the Distributive Property to multiply the binomials, the result is a 2 + ab + ab + b 2 before simplifying. Combining the two middle terms gives Module 18 678 Lesson 3 2ab. Collaborative Learning Peer-to-Peer Activity Avoid Common Errors Have students work in pairs. Instruct each student to write two binomial When students see a square of a binomial such multiplication problems: a square of a binomial and a product of a sum and as (3x + 2) , they sometimes square only the two 2 difference with the same terms. Have partners switch papers and find the products, terms, losing the x-term of the trinomial. To avoid this showing their work. Then have them check each other’s results. Have students error, have students rewrite the expression in the form repeat with additional problems. of the special products rule for (a + b) : 2 (3x + 2)2 = (3x)2 + (2)(3x)(2) + (2)2. 678 Special Products of Binomials  Multiplying (a - b) 2 Explain 2 2 In the Explore, you determined the square of a binomial difference, (a - b) = a 2 - 2ab + b 2. Because a 2 - 2ab + b 2 is the result of squaring the binomial (a - b), a 2 - 2ab + b 2 is also a perfect-square trinomial. Explain 2 2 Example 2 Multiply. Multiplying (a − b) (x - 5) 2  (6x - 1) 2 (a - b) 2 = a 2 - 2ab + b 2 (a - b) 2 = a 2 - 2ab + b 2 Questioning Strategies ( ) ( )( 1 ) + How is the product of (a − b) similar to and 2 2 (x - 5) 2 = x 2 - 2(x)(5) + 5 2 (6x - 1) 2 = 6 x -2 6 x 1 different from the product of (a + b) ? Both 2 = x 2 - 10x + 25 = 36 x 2 - 12 x + 1 products are trinomials. The first and last terms Reflect are the same in both products, but the sign of the middle term is negative for (a − b) and positive 2 7. Why is the last term of a perfect square trinomial always positive? for (a + b). 2 Because it is either the product of two positive numbers or the product of two Your Turn negative numbers. In either case, the product is positive. Multiply. 8. (4x - 3y) 2 9. (3 - x 2) 2 Integrate Mathematical Practices 16x 2 - 24xy + 9y 2 9 - 6x 2 + x 4 Focus on Critical Thinking MP.3 When finding the product of (a − b) , some 2 Explain 3 Multiplying (a + b) (a - b) students might believe that the middle term and the In the Explore, you determined the formula (a + b) (a - b) = a 2 - b 2. A binomial of the form a 2 - b 2 is called a difference of two squares. last term will both be negative numbers. Remind them that the last term comes from a number being Example 3 Multiply. squared, and thus will never be negative. (x + 6)(x - 6)  (x 2 + 2y)(x 2 - 2y) (a + b)(a - b) = a 2 - b 2 (a + b)(a - b) = a 2 - b 2 Avoid Common Errors ( ) ( ) 2 2 (x + 6)(x - 6) = x 2 - 6 2 © Houghton Mifflin Harcourt Publishing Company (x 2 + 2y)(x 2 - 2y) = x2 - 2y Students may forget to square all factors of each term = x 2 - 36 = 1 x 4 - 2 when squaring a binomial. Suggest that they circle 4 y the coefficient and any exponents in each term so Reflect that they do not overlook them as they apply the 10. Why does the product of a + b and a - b always include a minus sign? special product rule. Because b and -b have opposite signs, their product is negative. Your Turn 11. (7 + x)(7 - x) 49 - x 2 Explain 3 Multiplying (a + b)(a − b) Module 18 679 Lesson 3 Questioning Strategies What can you say about the sign of the last term in Differentiate Instruction the product of a sum and a difference? Explain. The Visual Cues sign of the last term is always negative because you are multiplying a positive term and a negative term. When students use the special product patterns, encourage them to (1) write the general pattern, (2) list which terms are represented by a and b in the general In a product of the form (a + b)(a − b), how many pattern, (3) substitute the terms into the pattern, and (4) simplify. terms do you expect in the answer? Explain. There For example, to find (3x2 + 4) , they would write: 2 are only two terms because the product of the inner terms and the product of the outer terms are Find (a + b) = a2 + 2ab + b2 for a = 3x2 and b = 4. 2 opposites, so their sum is zero. (3x2 + 4) = (3x2) + 2(3x2)(4) + (4) 2 2 2 = 9x4 + 24x2 + 16 Lesson 18.3 679  Explain 4 Modeling with Special Products Explain 4 Example 4 Write and simplify an expression to represent the situation. Design A designer adds a border with a uniform width to a square rug. The original side length of the rug is Modeling with Special Products (x - 5) feet. The side length of the entire rug including the original rug and the border is (x + 5) feet. What is the area of the border? Evaluate the area of the border if x = 10 feet. Questioning Strategies Analyze Information Identify the important information. Why is it helpful to draw a diagram when solving a The answer will be an expression that represents the area of the border. real-world problem involving length, area, or List the important information: volume? Labeling the dimensions in the diagram t
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The side length of the entire square area including the original rug and the border x+ 5 see how to use those expressions to find the is feet. solution. Formulate a Plan ( ). The total area of the rug plus the border in square feet 2 The area of the rug in square feet is x- 5 ( ). The area of the rug can be subtracted from the total area to find the area of the border. 2 Integrate Mathematical Practices is x+ 5 Focus on Technology Solve MP.5 When a real–world problem calls for evaluating a polynomial for a particular value Find the total area: Find the area of the rug: ( )( 5 ) + ( x )( 5 ) + ( 5 ) 2 2 of the variable, students can use a graphing calculator (x + 5) = 1 x + 2 x 2 2 5 (x - 5) 2 = 1 x 2 - 2 to check their calculations. = 1 x 2 + 10 x + 25 = 1 x 2 - 10 x + 25 Find the area of the border: Area of border = total area – area of rug Elaborate Area = 1 x 2 + 10 x + 25 - ( 1 x 2 - 10 x + 25 ) © Houghton Mifflin Harcourt Publishing Company = 1 x + 10 x + 25 - 1 x + 10 x - 25 2 2 = ( ) ( 1 x 2 - 1 x 2 + 10 x + 10 x + ) ( 25 - 25 ) Integrate Mathematical Practices = 0 x 2 + 20 x + 0 Focus on Patterns = 20 x MP.8 Point out that when you use special product The area of the border is 0 x 2 + 20 x + 0 = 20x square feet. patterns, a and b can be numbers, variables, or variable expressions. Module 18 680 Lesson 3 Language Support Connect Vocabulary Discuss with students the various ways that the word square is used in this lesson. To square a number or expression is to raise it to a power of 2, or multiply it by itself. A perfect square is a number that results from squaring a number. A binomial raised to the second power, such as (x + 3) or (x − 3) , equals a trinomial whose 2 2 first and third terms are perfect squares. The trinomial as a whole is called a perfect square trinomial. The product of a binomial and a similar binomial with the opposite sign in the middle, such as (x + 2)(x − 2), is a difference of two squares, x2 − 2 2 or x2 − 4. 680 Special Products of Binomials  Justify and Evaluate Suppose that x = 10. The rug is 5 feet by 5 feet, so its area is 25 square feet. The total area is Summarize The Lesson ( 10 + 5 ) = 2 Complete the graphic organizer with students to 225 square feet, so the area of the border is 225 - 25 = 200 square feet, summarize the rules for special products of binomials. which is 20 (10) when x = 10. So the answer makes sense. Add a third column for students to provide examples Because x is a length, x is positive, so -32x represents a negative number. Area cannot of each type of product. Reflect be negative. She probably subtracted the total area from the area of the rug instead of the other way around. 12. Critique Reasoning Estelle solved a problem just like the example, except that the value of b in the two Special Products expressions was 8. Her expression for the area of the border was –32x. How do you know that she made an error? What do you think her error might have been? perfect square Your Turn (a + b) 2 trinomial Write and simplify an expression. a 2 + 2ab + b 2 perfect square 13. A square patio has a side length of (x - 3) feet. It is surrounded by a flower garden with a uniform width. The side length of the entire square area including the patio and the flower garden is (x + 3) feet. Write an expression for the area of the flower garden. (a - b) 2 trinomial Area of flower garden = total area - area of patio = x 2 + 6x + 9 - (x 2 - 6x + 9) = 12x a 2 − 2ab + b 2 difference of two Elaborate 15. No; there is no way to rewrite a binomial squared as a difference of squares. (a + b)(a − b) squares 14. How can you use the formula for the square of a binomial sum to write a formula for the square of a a2 − b2 ( ) 2 binomial difference? (a - b)2 = a + (-b) = a 2 + 2a(-b) + (-b)2 = a 2 - 2ab + b 2 15. Can you use the formula for the square of a binomial sum to write a formula for a difference of squares? See above. 16. Essential Question Check-In Use one of the special product rules to describe in words how to find the coefficient of xy in the product (5x - 3y). Multiply the coefficient of x in the binomial and the 2 coefficient of y in the binomial. Then multiply the product by -2. The coefficient of xy in the product is -2(5)(3) = -30. Evaluate Evaluate: Homework and Practice © Houghton Mifflin Harcourt Publishing Company Multiply. t
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1SBDUJDF Level Concepts Practice and Skills 3. (6 + x 2) 2 36 + 12x 2 + x 4 4. (-x + 5) 2 x 2 - 10x + 25 Basic Explore N/A (x + 11) x + 22x + 121 (8x + 9y) 64x + 144xy + 81y 2 2 2 2 2 5. 6. Example 1 Exercises 1–6, odd 7. (x - 3) 2 x 2 - 6x + 9 8. (5x - 2) 2 25x 2 - 20x + 4 Example 2 Exercises 7–12, even Example 3 Exercises 13–18, odd (6x - 7y) (36x 2) - 84xy + 49y 2 10. (5 - x 2) 2 25 - 10x 2 + x 4 2 9. Example 4 Exercises 19–23 (5x - 4y) 25x 2 - 40xy + 16y 2 12. (7 - 2x 2) 2 49 - 28x 2 + 4x 4 2 11. H.O.T. Exercise 25 Average Explore N/A Example 1 Exercises 1–6, even Module 18 681 Lesson 3 Example 2 Exercises 7–12, odd Example 3 Exercises 13–18, even Example 4 Exercises 19–23 Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices H.O.T. Exercises 25–26 Advanced Explore N/A 1–18 1 Recall of Information MP.5 Using Tools Example 1 Exercises 5–6 19–20 1 Recall of Information MP.4 Modeling Example 2 Exercises 10–12, 24 21 2 Skills/Concepts MP.4 Modeling Example 3 Exercises 16–18, 23 Example 4 Exercises 19–23 22–23 2 Skills/Concepts MP.5 Using Tools H.O.T. Exercises 25–26 24 2 Skills/Concepts MP.3 Logic Real World Problems 25–26 2 Skills/Concepts MP.3 Logic Lesson 18.3 681  13. (x + 4)(x - 4) x 2 - 16 14. (x 2 + 6y)(x 2 - 6y) x 4 - 36y 2 Cognitive Strategies 15. (9 + x)(9 - x) 81 - x 2 16. (2x + 5)(2x - 5) 4x 2 - 25 Remind students that if they forget the formulas for special product patterns, they can always use the 17. (3x 2 + 8y)(3x 2 - 8y) 9x 4 - 64y 2 18. (7 + 3x)(7 - 3x) 49 - 9x 2 Distributive Property to find the product. Write and simplify an expression to represent the situation. 19. Design A square swimming pool is surrounded by a cement walkway with a uniform width. The swimming pool has a side length of (x - 2) feet. The side length Avoid Common Errors of the entire square area including the pool and the walkway is (x + 1) feet. Write an expression for the area of the walkway. Then find the area of the cement walkway When using the rules for special products, students when x = 7 feet. See below. often forget to square the coefficients of terms in the 20. This week Leo worked (x + 4) hours at a pizzeria. He is paid (x - 4) dollars per hour. Leo’s friend Frankie worked the same number of binomial. Suggest that students first write the hours, but he is paid (x - 2) dollars per hour. Write an expression for coefficient and variable with the exponent applied the total amount paid to the two workers. Then find the total amount if to each one, and then simplify. For example, to x = 12. find (3y − 5x2) , they would write 2 Total amount paid = Leo’s pay + Frankie’s pay 2 3 2y2 − 2(3y)(5x2) + 5 2(x2) , then simplify to = (x + 4)(x - 4) + (x + 4)(x - 2) 9y2 − 30x2y + 25x4. = 2x 2 + 2x - 24 When x = 12, the total amount paid to the two workers was $288. Online Resources t
Practice and Problem Solving (three forms) 21. Kyra is framing a square painting with side lengths of (x + 8) inches. The total area of t
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Success for English Learners 22. Geometry Circle A has a radius of (x + 4) units. A larger circle, B, has a radius Name ________________________________________ Date __________________ Class __________________ of (x + 5) units. Use the formula A = πr 2 to write an expression for the difference LESSON Special Products of Binomials in the areas of the circles. Leave your answer in terms of π. Then use 3.14 for π to 18-3 Practice and Problem Solving: A/B approximate to the nearest whole number the difference in the areas when x = 10. *NBHFT.PYJF
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*NBHFT Differece = Area of circle B - area of circle A = π(x + 5)(x + 5) - π(x + 4)(x + 4) = 2πx + 9π; Find the product. 1. ( x  2)2 2. (m  4)2 3. (3  a )2 ________________________ _______________________ ________________________ 91 square units. 4. (2 x  5)2 5. (8  y )2 6. (a  10)2 ________________________ _______________________ ________________________ 7. (b  3)2 8. (3 x  7)2 9. (6  3n )2 19. Area of walkway = Total area - area of pool = (x + 1) - (x - 2) = 6x - 3 ________________________ _______________________ ________________________ 2 2 10. ( x  3)( x  3) 11. (8  y )(8  y ) 12. ( x  6)( x  6) ________________________ _______________________ ________________________ When x = 7 feet, the area of the walkway is 39 square feet. 13. (5 x  2)(5 x  2) 14. (4  2y )(4  2y ) 15. (10 x  7 y )(10 x  7 y ) 21. Area of frame = Total Area - Area of painting = (2x - 6) - (x + 8) = 3x 2 - 40x - 28 ________________________ _______________________ ________________________ 2 2 Solve. 16. Write a simplified expression for each of the following. When x = 16, the area of the frame is 100 square inches, so the cost is $8. a. area of the large rectangle _____________________________________ b. area of the small rectangle _____________________________________ c. area of the shaded area _____________________________________ 17. The small rectangle is made larger by adding 2 units to the length and Module 18 682 Lesson 3 2 units to the width. a. What is the new area of the smaller rectangle? _____________________________________ b. What is the area of the new shaded area? _____________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 323 682 Special Products of Binomials  23. A square has sides with lengths of (x - 1) units. A rectangle has a length of x units and a width of (x - 2) units. Which statements about the situation are true? Select all that apply. b, c, and d are true. Auditory Cues a. The area of the square is (x - 1) square units. 2 Have students learn “verbal rules” for special products b. The area of the rectangle is x 2 - 2x square units. and repeat them aloud to help remember them. For 2 2 c. The area of the square is greater than the area of the rectangle. example, the square of a binomial (a + b) or (a − b) is: d. The value of x must be greater than 2. t
first term squared e. The difference in the areas is 2x - 1. t
plus (or minus) two times the product 24. Explain the Error Marco wrote the expression (2x - 7y) = 4x 2 - 49y 2. Explain 2 of the terms and correct his error. See below. t
plus last term squared. H.O.T. Focus on Higher Order Thinking Have students create a similar verbal rule for 25. Critical Thinking Use the FOIL method to justify each special product rule. (a + b)(a − b) = a2 − b2. a. (a + b) 2 a 2 + ab + ab + b 2 = a 2 + 2ab + b 2 b. (a - b) a 2 - ab - ab - b 2 = a 2 - 2ab + b 2 2 c. (a + b)(a - b) a 2 + ab - ab - b 2 = a 2 - b 2 Avoid Common Errors 26. Communicate Mathematical Ideas Explain how you can use the special product Students may confuse the square of a rules and the Distributive Property to write a general rule for (a - b). Then write 3 difference, (p − q)2, with the difference of two the rule. squares, p2 − q2. Remind them that the square of a First write (a - b) as (a - b)(a - b)(a - b), then use the Distributive Property 3 to write the product. binomial is always a trinomial. The product of two (a - b)3 = a 3 - 3a 2b + 3ab 2 - b 3 binomials can be a binomial only if one term “drops out” because like terms cancel, as it does when a sum is multiplied by a difference. 24. Marco may have confused a difference of squares and the square of a binomial difference. He wrote the difference of the squares of the terms instead of square of a binomial difference. The correct product is 4x 2 - 28xy + 49y 2. © Houghton Mifflin Harcourt Publishing Company Module 18 683 Lesson 3 Lesson 18.3 683  Lesson Performance Task Journal When building a square-shaped outdoor fireplace, Have students write a journal entry describing how to the ground needs to be replaced with stone for an additional two feet on each side. Write a polynomial use special product patterns to find the square of a for the area that needs to be excavated to create an x binomial and the product of a sum and a difference. by x fireplace. Design your ideal space for sitting around a fire pit and relaxing. Add furniture, flowerbeds, rock gardens, and any other desired features. Integrate Mathematical Practices Evaluate the polynomial for the size fireplace you are Focus on Modeling including. MP.4 Encourage students to draw and label a diagram for the situation in the Lesson Performance The length and width of the area to be excavated are both 2 + x + 2 or x + 4 feet. Task. Therefore, the area that needs to be excavated is x 2 + 8x + 16. The plans for the outdoor space and excavated area will vary with the individual, but the excavated area will be A(x) evaluated for x, where x is the length/width of x+4 the fire pit. 2 x 2 x 2 x+4 ª
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2 Questioning Strategies What type of special product represents the area of the outdoor fireplace? It is a perfect-square trinomial, the result of squaring a binomial. 1IBSS4IVUUFSTUPDL Module 18 684 Lesson 3 Extension Activity Lesson Performance Task Have students investigate how Pascal’s triangle, which 1 was developed by 17 th–century mathematician Blaise Scoring Rubric 1 2 1 Pascal, is related to products of binomials. Students Points Criteria should find that the coefficients of a binomial raised 1 3 3 1 2 Student correctly solves the problem and explains to the nth power match the nth row in the triangle. 1 4 6 4 1 his/her reasoning. For example, the fourth row of the triangle corresponds 1 5 10 10 5 1 1 Student shows good understanding of the to (a + b) = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4. 4 problem but does not fully solve or explain his/ her reasoning. 0 Student does not demonstrate understanding of the problem. 684 Special Products of Binomials