Consider the polynomial q(x)=kx4+3x3−2x+1, where k is a real constant.
(i) Obtain expressions, in terms of k, for the remainders when q(x) is divided by (a) (x−2) and (b) (x+1).
(ii) If the two remainders obtained in part (i) are equal, form the resulting equation in k and solve it.
(iii) For the value(s) of k obtained in part (ii), determine the common remainder mentioned above and state whether (x+1) is a factor of q(x).