Laurie s Notes. Overview of Section 7.6. (1x + 6)(2x + 1)

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1 Laurie s Notes Overview of Section 7.6 Introduction In this lesson, students factor trinomials of the form ax 2 + bx + c. In factoring trinomials, an common factor should be factored out first, leaving the resulting coefficients with fewer factors to work with. Students will recognize that the fewest combinations occur when both a and c are prime. When either a or c is composite, there are more combinations to check. When both a and c are composite, there are even more combinations to check. The lesson ends with a real-life problem where once the polnomial has been factored and set equal to zero, the Zero-Product Propert is used to solve the equation. Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations Teaching Strateg Factoring trinomials in which the leading coefficient is not equal to 1 can be challenging for some students. The struggle to hold several computations in their heads. It is helpful to have a structure that allows students to test out different combinations quickl. Example: If 2x x + 6 factors over the integers, then the binomial factors must be (x + )(2x + ). We also know that the factors of the constant term c are {1, 2, 3, 6}. The goal is to find the combination that gives a linear term of 13x. We could write out the four possible combinations: (x + 1)(2x + 6) 1x x = 8x (x + 6)(2x + 1) 1x x = 13x (x + 2)(2x + 3) 1x x = 7x (x + 3)(2x + 2) 1x x = 8x There is a smbolic wa of representing all four cases. For each set of brackets, sum the products along both solid lines; sum the products along the dotted lines. The sum that is 13 is highlighted. Write the binomials that give the same calculations when multiplied (1x + 6)(2x + 1) The list of possible factor combinations can be written verticall for students to check. This helps students who need a structure for listing the possible combinations. Pacing Suggestion Complete the exploration and discuss students observations. Transition to the formal lesson. Section 7.6 T-390

2 Common Core State Standards HSA-SSE.A.2 Use the structure of an expression to identif was to rewrite it. HSA-SSE.B.3a Factor a quadratic expression to reveal the zeros of the function it defines. Laurie s Notes Exploration Motivate Model 2x 2 + 3x + 1 with tiles on the overhead. Have students do the same at their desks. Can ou arrange the algebra tiles so the form a rectangle with no holes or overlaps? es When students tr to place all three x-tiles verticall, there is no place for the unit tile to fit. Model this when students do not make the attempt. What are the dimensions of the rectangle? (x + 1) and (2x + 1) Label the dimensions. Explain that 2x 2 + 3x + 1 = (x + 1)(2x + 1). MP5 Use Appropriate Tools Strategicall: This is similar to the visual model used when whole numbers are multiplied using base-ten blocks. Exploration 1 Ask students to form a rectangular arra to model 2x 2 + 5x + 2. The two x 2 -tiles need to be placed side b side, either verticall or horizontall. Show the following three arrangements and explain that we use the one on the left because there is space for the two remaining unit tiles. What are the dimensions of the rectangle? (x + 2) and (2x + 1) Label the dimensions. Explain that 2x 2 + 5x + 2 = (x + 2)(2x + 1). As students work part (a) with their partners, encourage them to think about the dimensions of the rectangle. Part (b) has been started so it eliminates the possibilit of (4x + )(x + ). Part (c) has onl one possibilit for the x 2 -tiles, though students need to pa attention to signs as the work the problem. Communicate Your Answer MP5: The algebra tiles provide a visual confirmation that the trinomial in Question 3 is not factorable. Connecting to Next Step The exploration helps students visualize and make sense of factoring a trinomial of the form ax 2 + bx + c. Once explored, students will be better able to perform the algebraic manipulations in the formal lesson. T-391 Chapter 7

3 7.6 Factoring ax 2 + bx + c Essential Question How can ou use algebra tiles to factor the trinomial ax 2 + bx + c into the product of two binomials? Finding Binomial Work with a partner. Use algebra tiles to write each polnomial as the product of two binomials. Check our answer b multipling. Sample 2x 2 + 5x + 2 Step 1 Arrange algebra tiles that Step 2 Use additional algebra tiles model 2x 2 + 5x + 2 into a to model the dimensions of rectangular arra. the rectangle. Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations ANSWERS 1. a. (x + 1)(3x + 2) b. (2x + 3)(2x 1) Step 3 Write the polnomial in factored form using the dimensions of the rectangle. width length Area = 2x 2 + 5x + 2 = (x + 2)(2x + 1) a. 3x 2 + 5x + 2 = c. (x 5)(2x 1) b. 4x 2 + 4x 3 = c. 2x 2 11x + 5 = USING TOOLS STRATEGICALLY To be proficient in math, ou need to consider the available tools, including concrete models, when solving a mathematical problem. Communicate Your Answer 2. How can ou use algebra tiles to factor the trinomial ax 2 + bx + c into the product of two binomials? 3. Is it possible to factor the trinomial 2x 2 + 2x + 1? Explain our reasoning. Section 7.6 Factoring ax 2 + bx + c Arrange algebra tiles that model the trinomial into a rectangular arra, use additional algebra tiles to model the dimensions of the rectangle, then write the polnomial in factored form using the dimensions of the rectangle. 3. no; There is no wa to model this expression as a rectangular arra. Section

4 Extra Example 1 Factor 4x x (x + 3)(x + 5) Extra Example 2 Factor each polnomial. a. 2x 2 + 7x + 6 (2x + 3)(x + 2) b. 4x 2 7x + 3 (4x 3)(x 1) 7.6 Lesson What You Will Learn Core Vocabular Previous polnomial greatest common factor (GCF) Zero-Product Propert Factor ax 2 + bx + c. Use factoring to solve real-life problems. Factoring ax 2 + bx + c In Section 7.5, ou factored polnomials of the form ax 2 + bx + c, where a = 1. To factor polnomials of the form ax 2 + bx + c, where a 1, first look for the GCF of the terms of the polnomial and then factor further if possible. Factoring Out the GCF Factor 5x x Notice that the GCF of the terms 5x 2, 15x, and 10 is 5. 5x x + 10 = 5(x 2 + 3x + 2) Factor out GCF. = 5(x + 1)(x + 2) Factor x 2 + 3x + 2. So, 5x x + 10 = 5(x + 1)(x + 2). When there is no GCF, consider the possible factors of a and c. Factoring ax 2 + bx + c When ac Is Positive STUDY TIP You must consider the order of the factors of 3, because the middle terms formed b the possible s are different. Factor each polnomial. a. 4x x + 3 b. 3x 2 7x + 2 a. There is no GCF, so ou need to consider the possible factors of a and c. Because b and c are both positive, the factors of c must be positive. Use a table to organize information about the factors of a and c. of 4 of 3 1, 4 1, 3 (x + 1)(4x + 3) 3x + 4x = 7x 1, 4 3, 1 (x + 3)(4x + 1) x + 12x = 13x 2, 2 1, 3 (2x + 1)(2x + 3) 6x + 2x = 8x So, 4x x + 3 = (x + 3)(4x + 1). b. There is no GCF, so ou need to consider the possible factors of a and c. Because b is negative and c is positive, both factors of c must be negative. Use a table to organize information about the factors of a and c. of 3 of 2 1, 3 1, 2 (x 1)(3x 2) 2x 3x = 5x 1, 3 2, 1 (x 2)(3x 1) x 6x = 7x So, 3x 2 7x + 2 = (x 2)(3x 1). 392 Chapter 7 Polnomial Equations and Factoring Laurie s Notes Teacher Actions Students now understand that the abilit to factor a trinomial is related to the values of a, b, and c. Factoring out the GCF will make sense! Turn and Talk and Popsicle Sticks: When a and c are positive, what does the sign of b tell ou? Use Popsicle Sticks to solicit an explanation. The more factors a and c have, the more combinations must be checked. See the Teaching Strateg on page T-390 for a structure to list and check the various combinations. 392 Chapter 7

5 STUDY TIP When a is negative, factor 1 from each term of ax 2 + bx + c. Then factor the resulting trinomial as in the previous examples. Factor 2x 2 5x 7. Factoring ax 2 + bx + c When ac Is Negative There is no GCF, so ou need to consider the possible factors of a and c. Because c is negative, the factors of c must have different signs. Use a table to organize information about the factors of a and c. of 2 of 7 1, 2 1, 7 (x + 1)(2x 7) 7x + 2x = 5x 1, 2 7, 1 (x + 7)(2x 1) x + 14x = 13x 1, 2 1, 7 (x 1)(2x + 7) 7x 2x = 5x 1, 2 7, 1 (x 7)(2x + 1) x 14x = 13x So, 2x 2 5x 7 = (x + 1)(2x 7). Factor 4x 2 8x + 5. Factoring ax 2 + bx + c When a Is Negative Step 1 Factor 1 from each term of the trinomial. 4x 2 8x + 5 = (4x 2 + 8x 5) Step 2 Factor the trinomial 4x 2 + 8x 5. Because c is negative, the factors of c must have different signs. Use a table to organize information about the factors of a and c. of 4 of 5 1, 4 1, 5 (x + 1)(4x 5) 5x + 4x = x 1, 4 5, 1 (x + 5)(4x 1) x + 20x = 19x 1, 4 1, 5 (x 1)(4x + 5) 5x 4x = x 1, 4 5, 1 (x 5)(4x + 1) x 20x = 19x 2, 2 1, 5 (2x + 1)(2x 5) 10x + 2x = 8x 2, 2 1, 5 (2x 1)(2x + 5) 10x 2x = 8x So, 4x 2 8x + 5 = (2x 1)(2x + 5). Monitoring Progress Factor the polnomial. Help in English and Spanish at BigIdeasMath.com 1. 8x 2 56x x x x 2 7x x 2 14x x 2 19x x 2 + x m 2 + 6m x 2 x + 2 Differentiated Instruction Organization Some students ma benefit from completing a table to show the signs of the factors of c. Students can use their tables to check the signs of the factors of c in their s. ax 2 + bx + c b c of c positive positive both positive positive negative different signs negative positive both negative negative negative different signs Extra Example 3 Factor 3x 2 7x 6. (x 3)(3x + 2) Extra Example 4 Factor 9x 2 3x + 2. (3x 1)(3x + 2) MONITORING PROGRESS ANSWERS 1. 8(x 1)(x 6) 2. (7x + 5)(2x + 3) 3. (x 1)(2x 5) 4. (3x 2)(x 4) 5. (4x + 1)(x 5) 6. (2x + 3)(3x 4) 7. ( + 1)(2 + 3) 8. (m 1)(5m 1) 9. (x + 1)(3x 2) Section 7.6 Factoring ax 2 + bx + c 393 Laurie s Notes Teacher Actions Write Example 3. Because the constant term is 7, what do ou know about the linear factors? The will have different signs. Some students will have little difficult using mental math to check the combinations. After modeling a problem, students must tr problems independentl. Watching someone factor trinomials cannot replicate the independent thought process that students need to experience. MP7 Look For and Make Use of Structure: In Example 4, factoring out a ( 1) is not an intuitive step for students. Make the connection to the Distributive Propert again. Section

6 Extra Example 5 The length of a rectangular state park is 2 miles longer than twice the width. The area of the park is 84 square miles. What is the width of the park? The width of the park is 6 miles. MONITORING PROGRESS ANSWER mi Solving Real-Life Problems Solving a Real-Life Problem The length of a rectangular game reserve is 1 mile longer than twice the width. The area of the reserve is 55 square miles. What is the width of the reserve? Use the formula for the area of a rectangle to write an equation for the area of the reserve. Let w represent the width. Then 2w + 1 represents the length. Solve for w. w(2w + 1) = 55 Area of the reserve 2w 2 + w = 55 Distributive Propert 2w 2 + w 55 = 0 Subtract 55 from each side. Factor the left side of the equation. There is no GCF, so ou need to consider the possible factors of a and c. Because c is negative, the factors of c must have different signs. Use a table to organize information about the factors of a and c. of 2 of 55 1, 2 1, 55 (w + 1)(2w 55) 55w + 2w = 53w 1, 2 55, 1 (w + 55)(2w 1) w + 110w = 109w 1, 2 1, 55 (w 1)(2w + 55) 55w 2w = 53w 1, 2 55, 1 (w 55)(2w + 1) w 110w = 109w 1, 2 5, 11 (w + 5)(2w 11) 11w + 10w = w 1, 2 11, 5 (w + 11)(2w 5) 5w + 22w = 17w 1, 2 5, 11 (w 5)(2w + 11) 11w 10w = w 1, 2 11, 5 (w 11)(2w + 5) 5w 22w = 17w Check Use mental math. The width is 5 miles, so the length is 5(2) + 1 = 11 miles and the area is 5(11) = 55 square miles. So, ou can rewrite 2w 2 + w 55 as (w 5)(2w + 11). Write the equation with the left side factored and continue solving for w. (w 5)(2w + 11) = 0 Rewrite equation with left side factored. w 5 = 0 or 2w + 11 = 0 Zero-Product Propert w = 5 or 11 w = Solve for w. 2 A negative width does not make sense, so ou should use the positive solution. So, the width of the reserve is 5 miles. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 10. WHAT IF? The area of the reserve is 136 square miles. How wide is the reserve? 394 Chapter 7 Polnomial Equations and Factoring Laurie s Notes Teacher Actions MP1 Make Sense of Problems and Persevere in Solving Them and MP5: Ask a student to read the problem, make a sketch, and label the dimensions of the rectangle. Once the trinomial is set equal to 0 ask, To get a sum of 1w for the middle term, what factors of 55 might be reasonable to tr first? Wh? 5 and 11, because the 5 can be multiplied b 2w first. Closure Exit Ticket: Factor 2x 2 7x + 3. (2x 1)(x 3) 394 Chapter 7

7 7.6 Exercises Dnamic Solutions available at BigIdeasMath.com In Exercises 3 8, factor the polnomial. (See Example 1.) 3. 3x 2 + 3x v 2 + 8v k k b 2 63b r 2 36r 45 In Exercises 9 16, factor the polnomial. (See Examples 2 and 3.) 9. 3h h m m x 2 5x w 2 31w n 2 + 5n z 2 + 4z g 2 10g v 2 15v 18 In Exercises 17 22, factor the polnomial. (See Example 4.) 17. 3t t v 2 25v c c h 2 13h w 2 w d d 9 ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in factoring the polnomial Vocabular and Core Concept Check 1. REASONING What is the greatest common factor of the terms of ? 2. WRITING Compare factoring 6x 2 x 2 with factoring x 2 x 2. Monitoring Progress and Modeling with Mathematics 2x 2 2x 24 = 2(x 2 2x 24) = 2(x 6)(x + 4) 6x 2 7x 3 = (3x 3)(2x + 1) In Exercises 25 28, solve the equation x 2 5x 30 = k 2 5k 18 = n 2 11n = b 2 2 = 3b 30. x = 3, x = x = 1, x = x = 1 3, x = a. (5x 2) ft b. Substitute 3 for x into the expression for the area 15x 2 x 2, then simplif; Substitute 3 for x into the expressions for the length (5x 2) and width (3x + 1), simplif each, then multipl these two numbers. In Exercises 29 32, find the x-coordinates of the points where the graph crosses the x-axis = 2x 2 3x = 7x 2 2x + 5 x x MODELING WITH MATHEMATICS The area (in square feet) of the school sign can be represented b 15x 2 x 2. a. Write an expression that represents the length of the sign. b. Describe two was to find the area of the sign when x = 3. (3x + 1) ft 1 x Section 7.6 Factoring ax 2 + bx + c = 4x x = 3x x + 5 x Assignment Guide and Homework Check ASSIGNMENT Basic: 1, 2, 3 25 odd, 33, 35, 38, 40, Average: 1, 2 40 even, Advanced: 1, 2, 8, even, HOMEWORK CHECK Basic: 3, 9, 13, 17, 35 Average: 4, 10, 14, 18, 34 Advanced: 8, 12, 16, 20, 36 ANSWERS Factoring 6x 2 x 2 requires considering factors of 6 and 2 in different combinations until the combination is found that produces the correct middle term. Factoring x 2 x 2 onl requires finding the factors of 2 that add up to (x 1)(x + 2) 4. 8(v 2)(v + 3) 5. 4(k + 3)(k + 4) 6. 6( 1)( 3) 7. 7(b 4)(b 5) 8. 9(r + 1)(r 5) 9. (3h + 2)(h + 3) 10. (2m + 7)(4m + 1) 11. (2x 1)(3x 1) 12. (2w 5)(5w 3) 13. (n + 2)(3n 1) 14. (2z 1)(2z + 3) 15. 2(g 2)(4g + 3) 16. 3(2v 3)(3v + 2) 17. (t 3)(3t 2) 18. (v + 3)(7v + 4) 19. (c 5)(4c + 1) 20. (h + 2)(8h 3) 21. (3w 4)(5w + 7) 22. (2d 1)(11d 9) 23. need to factor 2 out of ever term; = 2(x 2 x 12) = 2(x + 3)(x 4) 24. These factors do not give the correct middle term; = (2x 3)(3x + 1) 25. x = 2, x = k = 2, k = n = 5 3, n = b = 1 2, b = x = 7 2, x = 5 Section

8 Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations ANSWERS sec 35. length: 70 m, width: 31 m 36. es; The length of the invitation is 5 inches, which is less than 5 1 inches. The width of the 8 invitation is 3 inches, which is less than inches. 37. Sample answer: 6x 2 + 3x 38. The graph of k represents function g, and the graph of represents function h; Because c is positive, the constant terms in the factors must have the same sign. Because g has a positive value of b, the constant terms of the factors will both be positive, which results in negative roots, and k has two negative x-intercepts. Because h has a negative value of b, the constant terms of the factors will both be negative, which results in positive roots, and has two positive x-intercepts. 39. when no combination of factors of a and c produce the correct middle term; Sample answer: 2x 2 + x no; To use the Zero-Product Propert, one side of the equation needs to be 0. So, ou must first subtract 2 from each side of the equation, then factor. 41. ±9, ±12, ± See Additional Answers. 34. MODELING WITH MATHEMATICS The height h (in feet) above the water of a cliff diver is modeled b h = 16t 2 + 8t + 80, where t is the time (in seconds). How long is the diver in the air? 35. MODELING WITH MATHEMATICS The Parthenon in Athens, Greece, is an ancient structure that has a rectangular base. The length of the base of the Parthenon is 8 meters more than twice its width. The area of the base is about 2170 square meters. Find the length and width of the base. (See Example 5.) 36. MODELING WITH MATHEMATICS The length of a rectangular birthda part invitation is 1 inch less than twice its width. The area of the invitation is 15 square inches. Will the invitation fit in the envelope shown without being folded? Explain. 37. OPEN-ENDED Write a binomial whose terms have a GCF of 3x. 38. HOW DO YOU SEE IT? Without factoring, determine which of the graphs represents the function g(x) = 21x x + 12 and which represents the function h(x) = 21x 2 37x Explain our reasoning. 2 k in. 8 2 x 39. REASONING When is it not possible to factor ax 2 + bx + c, where a 1? Give an example. 5 3 in. 8 Maintaining Mathematical Proficienc 40. MAKING AN ARGUMENT Your friend sas that to solve the equation 5x 2 + x 4 = 2, ou should start b factoring the left side as (5x 4)(x + 1). Is our friend correct? Explain. 41. REASONING For what values of t can 2x 2 + tx + 10 be written as the product of two binomials? 42. THOUGHT PROVOKING Use algebra tiles to factor each polnomial modeled b the tiles. Show our work. a. b. 43. MATHEMATICAL CONNECTIONS The length of a rectangle is 1 inch more than twice its width. The value of the area of the rectangle (in square inches) is 5 more than the value of the perimeter of the rectangle (in inches). Find the width. 44. PROBLEM SOLVING A rectangular swimming pool is bordered b a concrete patio. The width of the patio is the same on ever side. The area of the surface of the pool is equal to the area of the patio. What is the width of the patio? 16 ft 24 ft In Exercises 45 48, factor the polnomial k 2 + 7jk 2j x 2 + 5x a ab 14b m m 2 n 15mn 2 Reviewing what ou learned in previous grades and lessons Find the square root(s). (Skills Review Handbook) 49. ± ± 81 Solve the sstem of linear equations b substitution. Check our solution. (Section 5.2) 53. = 3 + 7x 54. 2x = x 2 = x 8 = x = 3 x + 3 = 14 7 = 2x x = Chapter 7 Polnomial Equations and Factoring Mini-Assessment Factor each polnomial. 1. 2x x (x + 3)(x + 4) 2. 3x x + 10 (3x + 5)(x + 2) 3. 2x 2 7x 4 (2x + 1)(x 4) 4. 3x 2 4x + 7 (3x + 7)(x 1) 5. The length of a rectangular garden is 4 ards less than twice the width. The area of the garden is 96 square ards. What is the width of the garden? The width of the garden is 8 ards. If students need help... Resources b Chapter Practice A and Practice B Puzzle Time Student Journal Practice Differentiating the Lesson Skills Review Handbook If students got it... Resources b Chapter Enrichment and Extension Cumulative Review Start the next Section 396 Chapter 7