Algebra and up

I love math until it gets in to polynomials that add on top of each other, from slope intercept to the quadratic formula to Trig.  It's usually in Trig that the "what's the point of it all" instinct kicks in, and it ends horribly.  The basis is in polynomials and formulas using them, which starts in Algebra.

How does one get to enjoy polynomial equations?  How does one feel good about doing them?  I have no love for polynomials.  No love at all.

I just need some way to visualize what a polynomial "means."  I know that in some vague sense, it represents a line on a graph, but I don't see "how" just by looking at a polynomial.  It's gibberish to me.  I understand X = X coordinate, Y = Y coordinate, and somehow number next to a variable equals a slope.