My guess is that n has to be even. My (informal) thinking goes like this.

If you think about the graph of the equation, you can see what happens when x goes to .

When n is odd, it starts at when and goes to when . Somewhere it has to cross x=0 at least once.

However, if n is even then as , we have that your equation goes to . It starts bigly positive and ends bigly positive and you can always assign a value of k to make sure it misses x=0 altogether.

You ought to be able to bag that up into a proof in whatever form that the level of your course requires.