A polynomial on it's own does not have any solutions.

ax2 + bx + c

has no solutions, since it is not an equation. When we say the roots or solutions of ax2 + bx + c we are usually assuming it equals 0 and is therefore an equation.

For ALL real values of x, an nth degree polynomial (one which starts axn + ...) has n solutions when equal to 0. Some of them might be the same, for example:

x4 - 6x3 + 13x2 - 12x + 4 = 0

This has four solutions: x = 1, 1, 2 or 2. Yes, obviously some of them are the same, but it doesn't mean they're not there. This is what you'd expect from a polynomial of degree 4.

Interestingly enough, if the polynomial is of degree 1/n, it will still have n solutions (figure it out).

Geometrically speaking:

the polynomial ax2 + bx + c = 0 always has 2 solutions. If they are different, then the line y = 0 crosses the parabola at two places. If the solutions are both the same then the line y = d is a tangent to the parabola. If the solutions are complex (i.e. if b2 - 4ac is negative), the lines do not cross.

With the example above, two pairs of identical solutions means that the polynomial above is a tangent to the line y = 0 at 2 places , (1, 0) and (2, 0).

So to finish:
1) Any equation where xn is the highest power of x, has n roots (or 1/n roots if it's x1/n).
2) If the roots form in pairs, then geometrically you have a tangent.
3) So let's stop arguing about y = x0.5, and get on with our lives