Question

let P(x)=x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f
If f is a prime number, how many distinct linear factors with integral coefficients can P(x) at most have?

Thanks very much.

Answer 1

recall that the product of the roots will be f/1

If f is prime, then the only possible rational roots will be

±1, ±f

Since f and -f cannot both be roots (why?) there can be at most 3 rational roots.

Furthermore, since f is positive (why?), -1 cannot be a root, since then the polynomial will end with "-f" instead of f.

SO, only 1 and f can be roots.