76. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content


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1 76 Choosing a Factoring Model Extension: Factoring Polynomials with More Than One Variable Essential question: How can you factor polynomials with more than one variable? What is the connection between x  81 and m  n? They are both the difference of squares. Standards for Mathematical Content Factoring by GCF Factoring ax + bx + c Factoring special products In the perfect square trinomial 4x + 1xy + 9y, what represents a and what represents b? Math Background Students have factored polynomials using a variety of methods: by grouping, by using special product patterns, by using the factors of the first and third terms. All these polynomials involved a single variable. Now students will factor polynomials that involve more than one variable. They will discover that the methods they learned for single variable polynomials apply equally well to multivariable polynomials. x represents a; 3y represents b. EXTRA Factor the polynomial completely. Explain each step. 3x + 36xy  108y 3(x  1xy + 36y) Factor the GCF. 3(x  6y)(x  6y) Perfect square trinomial Review the methods students have learned to factor polynomials that involve a single variable. Give students an example of each type of polynomial and have them do the factoring. Then give them polynomials with two variables that can be factored using the special product patterns. Lead them to see that the patterns still apply. Ask what they think this means about methods for factoring multivariable polynomials in general. Avoid Common Errors Students may confuse the perfect square trinomial rule with the difference of squares rule and use a  b = (a  b)(a  b). To help students remember that a  b = (a  b)(a + b), have them remember the statement Two terms = Two signs. T EACH EXPLORE If the second polynomial were 3m  3, how would you factor it? Use the GCF and the difference of squares: 3(m  1)(m + 1). 397 Lesson 6 Teaching Strategies Have groups of students develop a stepbystep plan to factor polynomials. Ask them to make a list of questions they can ask themselves as they work. Such questions might include Does this polynomial follow the pattern of Difference of Squares? Does it follow the pattern of Perfect Squares? Is there a GCF? What are some factors of the third coefficient? What is the sum of these factors? Allow students to refer to this plan, as necessary, when they factor polynomials. IN T RO DUC E Chapter 7 How do you know which special product pattern to use when factoring a perfect square trinomial? Look at the first operation sign. If it is +, factor as (a + b)(a + b). If it is , factor as (a  b)(a  b). Prerequisites 1 How would you factor xy  rs? Use the difference of squares rule: (xy + rs)(xy  rs) ASSE.1.1b Interpret complicated expressions by viewing one or more of their parts as a single entity.* ASSE.1. Use the structure of an expression to identify ways to rewrite it.
2 Name Class Date Choosing a Factoring Method Extension: Factoring Polynomials with More Than One Variable Essential question: How can you factor polynomials with more than one variable? 76 Notes 1 ASSE.1. EXPLORE Factoring Polynomials with Two Variables Factor each polynomial completely. Explain each step. x  16 ( x  81) Factor the GCF. x(x + 9)(x  9) Difference of squares 3 m  3 n 3 ( m  n ) Factor the GCF. 3(m + n)(m  n) Difference of squares 1a. Compare the methods of factoring the two polynomials. The methods are the same, except the second polynomial has two variables. 1b. How would the factoring change for the polynomial 3 m  1 n? After factoring the GCF, you would factor m  4 n, which is still a difference of squares. To factor a perfect square trinomial with more than variable you can use the same patterns you used with perfect square trinomials in one variable. ASSE.1. a + ab + b = (a + b)(a + b) a  ab + b = (a  b)(a  b) Factoring a Perfect Square Trinomial Factor the trinomial completely. Explain each step. 3 g + 1gh + 1 h 3 ( g + 4gh + 4 h ) Factor the GCF. 3(g + h)(g + h) Perfect square trinomial pattern Chapter Lesson 6 a. Which pattern would you use to factor 16 a  48ab + 36 b? Explain how you know and then factor the trinomial. a  ab + b = (a  b)(a  b); the first operation sign is ; 4(a  3b)(a  3b) 3 If there is no obvious pattern shown by the trinomial, you can find the factors of the coefficient of the third term and check their sums to find how to factor. ASSE.1. Factors of 1 1 and 1 and and Factoring a Polynomial Factor 3 x + 1xy + 36 y completely. Explain your steps. 3 ( x + 7xy + 1 y ) Find the factors of 1 that add to (x + 3y)(x + 4y) Sum of factors Factor the GCF. Factor according to the sum. 3a. Would finding the factors and their sum be enough to factor the polynomial x + 11xy + 1 y? Why or why not? 3b. What additional conditions must you consider to factor x + 11xy + 1 y? No, because the xterm of the binomial has a coefficient of. This would affect the middle term of the polynomial, so you can t just find the factors and the sum. You have to consider that one factor of 1 has been doubled. You will have to find which factor by trying to divide each factor by, then finding the sum. Sometimes grouping will allow you to factor a polynomial. Chapter Lesson 6 Chapter Lesson 6
3 3 What tells you that this polynomial is not a perfect square polynomial? 36 is a perfect square, but the coefficient of the first term, 3, is not. Describe the relationship between the coefficient of the second term and the coefficient of the third term after the GCF has been factored out. The coefficient of the second term is the sum of the two factors of the coefficient of the third term. The coefficient of the third term is the product of these two factors. EXTRA Factor m  8md + 15d completely. E xplain your steps. Does not have a common GCF. Does not follow any special pattern. Multiply the coefficients of the first and third terms: 1 15 = 15. The product is positive, so the factors are either both positive or both negative. The middle term is negative, so both factors will be negative. Find the negative factors of 15 and their sums. Factors Sum 1 and and 58 The sum of 3 and 5 is 8, the coefficient of the middle term. Factor according to the sum: (m  3d)(m  5d) 4 Suppose the polynomial were 7x 37xy + 6y  6x. What would be the first step of the factoring? Factor out the common GCF of 3. Do you have to factor 6x + 1xy + 3x + 4y by grouping? Explain. No; you can combine the like terms 6x and 3x to get 9x + 1xy + 4y which is a perfect square trinomial. EXTRA Factor x 3 + x y + x + y completely. Explain your steps. Does not have a common GCF. Group terms with a common factor: (x 3 + x y) + (x + y) Factor out the GCF of each group: x (x + y) + 1(x + y) Factor out the GCF of the products: (x + 1)(x + y) CLOSE Essential Question How can you factor polynomials with more than one variable? You can use the same methods you used for polynomials of one variable: difference of squares or perfect square trinomial patterns, sum of the factors method, or factoring by grouping. Summarize Have students write a journal entry describing the aspect of factoring they find most difficult. They should also note any reminders or hints they developed to help overcome this difficulty. PRACTICE Where skills are taught Where skills are practiced 1 EXPLORE EXS. 3, 6, 8 EXS. 1, 11, 15 3 EXS., 4, 5, 7, 9, 10, 1, 16 4 EXS. 13, 14 Chapter Lesson 6
4 4 ASSE.1.1b Factoring a Polynomial Using Grouping Notes Factor 9 x 39xy + y  x completely. Explain your steps. There is no GCF for all the terms. Since the polynomial has four terms, factor by grouping. ( 9 x 39xy ) + ( y  x ) Group terms that have a common 9x 9x 9x factor. The common factor of 9 x 3 y and 9xy is 9x. The common factor of y and  x is. ( x  y ) + ( ) ( y  x ) Factor out the GCF of each group. ( x  y ) + ( ) ( ) ( ) The polynomial contains the binomial ( x  y )  ( ) ( ) Simplify. 1 x  y x  y opposites ( x  y) and ( ). Write (y  x ) as (1) ( ). ( 9x  ) ( x  y ) Factor out ( x  y ), the common factor of the products. y  x x  y 4a. Describe the steps you would use in factoring 8 x  x + 4xy  6y. First I would factor out, the GCF of all the terms. I would then follow the steps in the Example to complete the factoring, although in this case there are no opposite binomials. PRACTICE Choose a factoring method to factor each polynomial completely. Explain each step. 1. x + 6xy + 9 x (x + 3y)(x + 3y) Perfect square trinomial. 4 x  4xy  8 y 4( x  xy  y ) Factor GCF. 4(x + y)(x  y) Factor according to sum. 3. x  4 y (x + y)(x  y) Difference of squares 4. g + 3gh  10 h (g  h)(g + 5h) Factor according to sum. Chapter Lesson 6 Factor each polynomial completely. 5. m + 5mn  3 n 6. 4 x  9 y (m + 3n)(m  n) (x + 3y)(x  3y) 7. g  7gh + 10 h b  49 c (g  5h)(g  h) 9. a 33 a b  4a b a + 3ab  18 b a( a  3ab  4 b ) a(a + b)(a  4b) 11. t t w + 18t w 1. 6 c 37 c d + 1c d t ( t + 6tw + 9 w ) t (t + 3w)(t + 3w) 13. x 3 y  5 x y + 4x x 4 y + 4 x y  7 x Factor x 4  y 4 completely. (Hint: What special form does this polynomial appear to follow?) ( x + y )( x  y ) ( x + y )(x + y)(x  y) 16. Jaime and Sam both factored the polynomial x + 10xy + 8 y. Which student is correct? Explain. Jaime Sam x + 10xy + 8 y x + 10xy + 8 y ( x + 5xy + 4 y ) (x + y)(x + 4y) (x + y)(x + 4y) (4b + 7c)(4b  7c) Jaime is correct. Sam did not completely factor the polynomial. The term (x + y) can be further factored to (x + y). 3( a + ab  6 b ) 3(a + b)(a  3b) 3c ( c  9cd + 4 d ) 3c (c  4d )(c  d ) x y(x  5) + 4(x  5) x y ( x + 4)  7( x + 4) ( x y + 4)(x  5) ( x y  7)( x + 4) Chapter Lesson 6 Chapter Lesson 6
5 ADDITIONAL PRACTICE AND PROBLEM SOLVING Assign these pages to help your students practice and apply important lesson concepts. For additional exercises, see the Student Edition. Answers Additional Practice 1. yes. no; 5m(m + 9) 3. no; p(p + 3)(p 3) 4. yes 5. no; 3jk 3 (5k + 19) 6. no; 14(7g 4 g + 5) 7. 8y(3xy + 5) 8. 5r(r s) 9. x y(3x + y) 10. 3b (a ) 11. (5t + 3)(t 3s)(t + 3s) 1. (y + 4x)(y 7x) 13. 3a(a + 3)(a + 5) 14. xy(x 3y)(x + 3y) 15. 1(n 3 4) 16. 3c (c + 4d) 17. not factorable w (w + 4 v )(w v)(w + v) Problem Solving 1. 3(5x 4y)(x + y). 4π(k 1) ; 4π m 3. 4(x + y)(x y) 4. 5(x 19)(x + 3); 55 attendees 5. D 6. G 7. C 8. J Chapter Lesson 6
6 Name Class Date Additional Practice 76 Notes Chapter Lesson 6 Problem Solving Chapter 7 40 Lesson 6 Chapter 7 40 Lesson 6
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