2. Factor each trinomial. Verify by multiplying the factors. a) 5c2 13cd + 6d2 – b) 3p2 5pq 2q2 – – Solution 5c2 13cd + 6d2 a) – Use logical reasoning. Since the 3rd term in the trinomial is positive, the operations in the binomial factors will be the same. Since the 2nd term is negative, these operations will be subtraction. The two binomials will have the form: (?c ?d)(?c ?d) – – List the possible factors of 5c2 and 6d2, then use guess and test to determine which combination of products has the sum of –13cd. 1c 1c 1c –1d –6d –2d 5c 5c 5c –6d –1d –3d 5cd = 30cd = 10cd = –6cd – –11cd –1cd – –31cd –3cd – –13cd This is the correct combination. So, 5c2 13cd + 6d2 = (c 2d)(5c 3d) – – – To verify, multiply: (c 2d)(5c 3d) = 3d) 2d(5c 3d) c(5c – – – – – = 5c2 3cd 10cd + 6d2 – – = 5c2 13cd + 6d2 – Since this trinomial is the same as the original trinomial, the factors are correct. (Solution continues.)
2. Factor each trinomial. Verify by multiplying the factors. a) 5c2 – 13cd + 6d2 b) 3p2 – 5pq – 2q2 Solution a) 5c2 – 13cd + 6d2 Use logical reasoning. Since the 3rd term in the trinomial is positive, the operations in the binomial factors will be the same. Since the 2nd term is negative, these operations will be subtraction. The two binomials will have the form: (?c – ?d)(?c – ?d) List the possible factors of 5c2 and 6d2, then use guess and test to determine which combination of products has the sum of –13cd. 1c –1d 1c –6d 1c –2d 5c –6d 5c –1d 5c –3d –6cd – 5cd = –11cd –1cd – 30cd = –31cd –3cd – 10cd = –13cd This is the correct combination. So, 5c2 – 13cd + 6d2 = (c – 2d)(5c – 3d) To verify, multiply: (c – 2d)(5c – 3d) = c(5c – 3d) – 2d(5c – 3d) = 5c2 – 3cd – 10cd + 6d2 = 5c2 – 13cd + 6d2 Since this trinomial is the same as the original trinomial, the factors are correct. (Solution continues.)