Polynomials and rational expressions are used to express properties of the physical world, like how gravity works. If you need a funnier example of gravity at work (or not at work), check this out.
Polynomials can also be used to do approximation, something that's usually covered in Calculus classes. Shudder. If you go here, you'll see polynomials whose shape become closer to what we want as we let the degree increase. These approximating polynomials are called Taylor polynomials, and they're used for everything from making your calculator work to storing music on CDs. You could even store a Taylor Swift CD using Taylor polynomials. We'd keep going, but it's probably a better idea not to.