Find all finite polynomials whose coefficients are all ±1 and whose roots are all real.

**Solution**

Answer: ±(x + 1), ±(x - 1), ±(x^{2} + x - 1), ±(x^{2} - x - 1), ±(x^{3} + x^{2} - x - 1), ±(x^{3} - x^{2} - x + 1).

The linear and quadratic polynomials are easy to find (and it is easy to show that they are the only ones).

Suppose the polynomial has degree n ≥ 3. Let the roots be k_{i}. Then ∑ k_{i}^{2} = (∑ k_{i})^{2} - 2 ∑ k_{i}k_{j} = a^{2} ± 2b, where a is the coefficient of x^{n-1} and b is the coefficient of x^{n-2}. Hence the arithmetic mean of the squares of the roots is (a^{2} ± 2b)/n. But it is at least as big as the geometric mean which is 1 (because it is an even power of c, the constant term). So we cannot have n > 3. For n = 3, we must have b the opposite sign to the coefficient of x^{n} and it is then easy to check that the only possibilities are those given above.

© John Scholes

jscholes@kalva.demon.co.uk

14 Jan 2002