Problem

Show that for all natural numbers k, the function

f(x) = x^4k - x^2k+1 + x^2k - x + 1

is always positive.

We break this into three cases: x < 1, x = 1 and x > 1.

When x < 1, factor the last four terms of f(x), so we can write f(x) = x^4k + (x^2k + 1)(1 - x); then each term is positive, and so f(x) > 0.
When x = 1, we can just plug in x to see f(1) = 1 > 0.
When x > 1, factor the first four terms of f(x), to write f(x) = x(x^2k-1 - 1)(x^2k + 1) + 1. Note that x^2k-1 > x > 1, and so every term is positive, and f(x) > 0, as desired.