a and b are unequal reals. What is the remainder when the polynomial p(x) is divided (x - a)^{2}(x - b).

**Solution**

Suppose the remainder is cx^{2} + dx + e. We have p(a) = ca^{2} + da + e, p(b) = cb^{2} + db + e. Also, differentiating, we get p'(a) = 2ca + d. Solving, c = p'(a)/(a - b) - p(a)/(a - b)^{2} + p(b)/(a - b)^{2}, d = (2a p(a) - 2a p(b) - (a^{2} - b^{2})p'(a) )/(a - b)^{2}, e = p(a) - a^{2}(p(a) - p(b))/(a - b)^{2} + ab p'(a) /(a - b).

© John Scholes

jscholes@kalva.demon.co.uk

5 Mar 2002