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(x2 - 1)(x2 - 4) = x5 - 5x3 + 4x. So we have to show that two coefficients are not possible.
The polynomial cannot be x5 + axn with n > 1, because it would then have 0 as a repeated root and hence less than 5 distinct roots. x5 + ax has at most 3 real roots, and x5 + a has only one real root.
© John Scholes
jscholes@kalva.demon.co.uk
7 Jan 2001