Let k be a non-zero integer. Prove that the polynomial p(x) = x^3 + x + k^2 has no rational roots as follows: (a) Show that any rational root must be...

Let k be a non-zero integer. Prove that the polynomial p(x) = x^3 + x + k^2 has no rational roots as follows:

(a) Show that any rational root must be an integer.

(b) Show that any integer root m of p(x) must be a perfect square.

(c) Assuming (b), conclude that p(x) has no rational roots.