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772Chapter 12 Polynomials: Factoring34. 125a 3 − 27b 3 35. 1 _8 x 3 − 1 _27 y 3 36. 1 _27 x 3 + 1 _8 y 337. a 6 − b 6 38. x 6 − 64y 6 39. 64x 6 − y 640. x 6 − (3y) 6 41. x 6 − (5y) 6 42. (4x) 6 − (7y) 6Getting Ready for the Next SectionMultiply each of the following.43. 2x 3 (x + 2)(x − 2) 44. 3x 2 (x + 3)(x − 3) 45. 3x 2 (x − 3) 2 46. 2x 3 (x + 5) 247. y(y 2 + 25) 48. y 3 (y 2 + 36) 49. (5a − 2)(3a + 1) 50. (3a − 4)(2a − 1)51. 4x 2 (x − 5)(x + 2) 52. 6x(x − 4)(x + 2) 53. 2ab 3 (b 2 − 4b + 1) 54. 2a 3 b(a 2 + 3a + 1)Maintaining Your Skills55. U.S. Energy The pie chart shows where Americans gettheir energy. In 2006, Americans used 101.6 quadrillionBtu of energy. How many Btu did Americans use fromPetroleum?56. Gas Prices The chart shows the breakdown in gas prices.If gasoline costs $4.05, how much is paid for the actualoil? Round to the nearest cent.Role of Renewable Energy In the U.S.Natural Gas 23%Coal 23%Renewable Energy 7%Nuclear Energy 8%Petroleum 40%Source: Energy Information Adminstration 2006What we pay for in a gallon of gasoline10%5%10%75%10%5%21%64%GasolineDieselSource: Energy Information Administration 2008TaxesDistributing & MarketingRefiningCrude OilSolve each equation for x.57. 2x − 6y = 8 58. −3x + 9y = 12 59. 4x − 6y = 8 60. −20x + 15y = −10Solve each equation for y.61. 3x − 6y = −18 62. −3x + 9y = −18 63. 4x − 6y = 24 64. −20x + 5y = −10
Factoring: A General Review 12.6In this section we will review the different methods of factoring that we presentedpreviously. Prior to this section, the polynomials you worked with were groupedtogether according to the method used to factor them; for example, in Section6.4 all the polynomials you factored were either the difference of two squares orperfect square trinomials. What usually happens in a situation like this is that youbecome proficient at factoring the kind of polynomial you are working with at thetime but have trouble when given a variety of polynomials to factor.We begin this section with a checklist that can be used in factoring polynomialsof any type. When you have finished this section and the problem set that follows,you want to be proficient enough at factoring that the checklist is second nature.ObjectivesA Factor a variety of polynomials.Examples now playing atMathTV.com/booksStrategyFactoring a PolynomialStep 1: If the polynomial has a greatest common factor other than 1, thenfactor out the greatest common factor.Step 2: If the polynomial has two terms (it is a binomial), then see if it is thedifference of two squares or the sum or difference of two cubes, andthen factor accordingly. Remember, if it is the sum of two squares, itwill not factor.Step 3: If the polynomial has three terms (a trinomial), then either it isa perfect square trinomial, which will factor into the square of abinomial, or it is not a perfect square trinomial, in which case youuse the trial and error method developed in Section 6.3.Step 4: If the polynomial has more than three terms, try to factor it bygrouping.Step 5: As a final check, see if any of the factors you have written can befactored further. If you have overlooked a common factor, you cancatch it here.AFactoring a Variety of PolynomialsHere are some examples illustrating how we use the checklist.Example 1Factor 2x 5 − 8x 3 .1. Factor 3x 4 − 27x 2 .Practice ProblemsSolution First, we check to see if the greatest common factor is other than 1.Since the greatest common factor is 2x 3 , we begin by factoring it out. Once we havedone so, we notice that the binomial that remains is the difference of two squares:2 x 5 − 8x 3 = 2x 3 (x 2 − 4) Factor out the greatest common factor, 2x 3= 2x 3 (x + 2)(x − 2) Factor the difference of two squaresNote that the greatest common factor 2x 3 that we factored from each term inthe first step of Example 1 remains as part of the answer to the problem; that isbecause it is one of the factors of the original binomial. Remember, the expressionwe end up with when factoring must be equal to the expression we startwith. We can’t just drop a factor and expect the resulting expression to equal theoriginal expression.Answer1. 3x 2 (x + 3)(x − 3)12.6 Factoring: A General Review773
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