(i) Show, for all a and b and for all j > 1, that ai - bi (a - b)(a)-1 +aj-26 +aj-362 + ... + abj-2 ... + abi-2 + b)-1). (ii) Let f(x) = Cpx"" +Cn-1x*-1 +...+cix+co be a polynomial of degree n. x"" For any number u, show there is some polynomial q(x) of degreen - 1 with .n- , f (x) = (x – u)q(x) + f(u). [Hint: n n n f (x) – f(u) = cixi - cju' = c;(x) – u'); -Σ j=0 j=0 now use part (i).] (iii) Show that u is a root of f(x) if and only if x -- u is a factor of f(x). f a