I am trying to understand the solution of the problem 4. https://www.imc-math.org.uk/imc2001/prob_sol1.pdf
My question is, $q(x) = 1+x+....+x^n$ is reducible on the complex field, so why can't $r(x)$ be expressed as the product of two polynomials with complex coefficients?, one of the them a factor of $q(x)$, so that we can have $r(x)$ with some roots of $q(x)$, but not all of them.
Thanks in advance!