Polynomials: Factoring - XYZ Custom Plus

744Chapter 12 Polynomials: Factoring9. Factor by grouping:5ax − 3ay + 10x − 6yExample 9Factor by grouping: 3ax − 2a + 15x − 10.Solution First, we factor a from the first two terms and 5 from the last twoterms. Then, we factor 3x − 2 from the remaining two expressions:3ax − 2a + 15x − 10 = a(3x − 2) + 5(3x − 2)= (3x − 2)(a + 5)Again, multiplying (3x − 2) and (a + 5) will convince you that these are the correctfactors.10. Factor completely:8x 2 − 12x + 10x − 15Example 10Factor 6x 2 − 3x − 4x + 2 by grouping.Solution The first two terms have 3x in common, and the last two terms haveeither a 2 or a −2 in common. Suppose we factor 3x from the first two terms and2 from the last two terms. We get:6x 2 − 3x − 4x + 2 = 3x (2x − 1) + 2(−2x + 1)We can’t go any further because there is no common factor that will allow us tofactor further. However, if we factor −2, instead of 2, from the last two terms, ourproblem is solved:6x 2 − 3x − 4x + 2 = 3x (2x − 1) − 2(2x − 1)= (2x − 1)(3x − 2)In this case, factoring −2 from the last two terms gives us an expression that canbe factored further.11. Factor:3x 2 + 7bx − 3xy − 7byExample 11Factor 2x 2 + 5ax − 2xy − 5ay.Solution From the first two terms we factor x. From the second two termswe must factor −y so that the binomial that remains after we do so matches thebinomial produced by the first two terms:2x 2 + 5ax − 2xy − 5ay = x (2x + 5a) − y (2x + 5a)= (2x + 5a)(x − y)Another way to accomplish the same result is to use the commutative property tointerchange the middle two terms, and then factor by grouping:2x 2 + 5ax − 2xy − 5ay = 2x 2 − 2xy + 5ax − 5ay= 2x (x − y) + 5a(x − y)= (x − y)(2x + 5a)This is the same result we obtained previously.CommutativepropertyGetting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.Answers9. (5x − 3y)(a + 2)10. (2x − 3)(4x + 5)11. (3x + 7b)(x − y)1. What is the greatest common factor for a polynomial?2. After factoring a polynomial, how can you check your result?3. When would you try to factor by grouping?4. What is the relationship between multiplication and factoring?