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 cx2 + dx + e. We have p(a) = ca2 + da + e, p(b) = cb2 + 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) - (a2 - b2)p'(a) )/(a - b)2, e = p(a) - a2(p(a) - p(b))/(a - b)2 + ab p'(a) /(a - b).
© John Scholes
jscholes@kalva.demon.co.uk
5 Mar 2002