21st Swedish 1981 Problem 3 Find all polynomials p(x) of degree 5 such that p(x) + 1 is divisible by (x-1) 3 and p(x) - 1 is divisible by (x+1) 3 .
p'(x) has a factor (x-1) 2 and a factor (x+1) 2 , but it has degree 4, so p'(x) = a(x 2 -1) 2 = ax 4 - 2ax 2 + a. Integrating, p(x) = ax 5 /5 - 2ax 3 /3 + ax + b. We have p(1) = -1, so a/5 - 2a/3 + a + b = -1, or 8a + 15b = -15. Also p(-1) = 1, so -a/5 + 2a/3 - a + b = 1, or -8a + 15b = 15. Hence a = -15/8, b = 0 and p(x) = (-3x 5 + 10x 3 - 15x)/8 Thanks to Suat Namli 21st Swedish 1981 © John Scholes jscholes@kalva.demon.co.uk 27 February 2004 Last corrected/updated 27 Feb 04