A polynomial p(x) is to be divided by a linear factor of the form (x−a).
(i) Express p(x) in the form (x−a)q(x)+r, where q(x) is a polynomial and r is a constant.
(ii) Substitute x=a in the expression obtained in part (i) and hence explain how the constant r can be obtained directly from p(x).
(iii) Using the reasoning of part (ii), describe a quick test to decide whether (x−4) is a factor of a given polynomial p(x) without performing the full division process.