46th Putnam 1985 Problem B1 p(x) is a polynomial of degree 5 with 5 distinct integral roots. What is the smallest number of non-zero coefficients it can have? Give a possible set of roots for a polynomial achieving this minimum. Solution
: 3. For example: 0, ±1, ±2. The example given has three non-zero coefficients: x(x 2 - 1)(x 2 - 4) = x 5 - 5x 3 + 4x. So we have to show that two coefficients are not possible. The polynomial cannot be x 5 + ax n with n > 1, because it would then have 0 as a repeated root and hence less than 5 distinct roots. x 5 + ax has at most 3 real roots, and x 5 + a has only one real root. 46th Putnam 1985 © John Scholes jscholes@kalva.demon.co.uk 7 Jan 2001