2. Factor each trinomial. a) 8x + 7 x2 – b) + 7a 18 a2 – Solution a) 8x + 7 x2 – Since factoring and expanding are inverse processes, there will be two binomial factors. They will have the form: (x + an integer)(x + an integer) The two integers in the binomial have a sum of and a product of 7. –8 To determine the integers, list of factors of 7, then add the factors in each pair. Since the product is positive, the factors have the same sign. Factors of 7 Sum of Factors 7 = –1, –7 –1 – –8 1, 7 1 + 7 = 8 The factors that have a sum of are and –8 –1 –7. Write these integers as the second terms in the binomials. 8x + 7 = (x 1)(x 7) x2 – – – b) + 7a 18 a2 – Find two integers whose sum is 7 and whose product is –18. Since the product is negative, the factors have different signs. Factors of Sum of Factors –18 18 + 18 = 17 –1, –1 1, 1 18 = –18 – –17 9 + 9 = 7 –2, –2 2, 2 9 = –9 – –7 6 + 6 = 3 –3, –3 3, 3 6 = –6 – –3 The factors that have a sum of 7 are and 9. –2 + 7a 18 = (a 2)(a + 9) a2 – –
2. Factor each trinomial. a) x2 – 8x + 7 b) a2 + 7a – 18 Solution a) x2 – 8x + 7 Since factoring and expanding are inverse processes, there will be two binomial factors. They will have the form: (x + an integer)(x + an integer) The two integers in the binomial have a sum of –8 and a product of 7. To determine the integers, list of factors of 7, then add the factors in each pair. Since the product is positive, the factors have the same sign. Factors of 7 Sum of Factors –1, –7 –1 – 7 = –8 1, 7 1 + 7 = 8 The factors that have a sum of –8 are –1 and –7. Write these integers as the second terms in the binomials. x2 – 8x + 7 = (x – 1)(x – 7) b) a2 + 7a – 18 Find two integers whose sum is 7 and whose product is –18. Since the product is negative, the factors have different signs. Factors of –18 Sum of Factors –1, 18 –1 + 18 = 17 1, –18 1 – 18 = –17 –2, 9 –2 + 9 = 7 2, –9 2 – 9 = –7 –3, 6 –3 + 6 = 3 3, –6 3 – 6 = –3 The factors that have a sum of 7 are –2 and 9. a2 + 7a – 18 = (a – 2)(a + 9)
miap_ch3_notes.docx