### 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**

Answer: 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