(a) Let f(x) = anxn ++ a1x + a0 Z[x]. If r/s Q, with gcd(r,...

Question:

(a) Let f(x) = anxn +∙∙∙∙∙∙∙∙+ a1x + a0 ∈ Z[x]. If r/s ∈ Q, with gcd(r, s) = land/(r/s) = 0, prove that s|an and r|a0.
(b) Find the rational roots, if any exist, of the following polynomials over Q. Factor f(x) in Q[x].
i) f(x) = 2x3 + 3x2 - 2x - 3
ii) f(x) = x4 + x3 - x2 - 2x - 2
(c) Show that the polynomial f(x) = x100 - x50 + x20 + x3 1 has no rational root.