2. Use the Factor Theorem to determine whether g(x) is a fac tor of p(x) in each of the following cases: (i) p(:r) = 23!1 + x2 - 2x - 1, g(x) = :r + 1 (ii) p(x) r3 + 3z2 + 3x + 1, g(:r) = :r + 2 (iii) p(x) = r3 - 4x2 + x + 6, g(:r) = x - 3.

(i) Ifg(:r) is a factor of p(x). then p(- 1) = 0. Now, p(- 1) • - 2 + 1 + 2 - 1 = 0. Hence, g(:r) is a factor of p(%). (ii") Ifg(x) is a factor of p(x), then p(- 2) = 0. Now, p(- 2) = -.8 + 12 - 6 + 1= - 14 + 13 = -1¢ 0. Hence, g(:r) is not a factor of p(.%). (iii") Ifg(:r) is a factor of p{x), then p(3)' = 0. Now, p(3) = 27 - 36 + 3 + 6 = 36 - 36 = 0. Hence, g(:r) is a factor of p(x).