Polynomials:

They exist because they are efficient to compute. Computers do well with basic arithmetic operations like addition (+) and multiplication (*). The polynomial functions are simply those that you can construct from those two operations, and constant numbers.

Like consider a polynomial like f(x) = 5x^3 + 3x^2 + 2x + 7

What it really says is f(x) = 5*x*x*x + 3*x*x + 2*x + 7 and here you can see how it's all built from + and *.

This is why polynomials are useful. Because computers have an easy time calculating them. And all modern mathematics is done on computers. All the engineering uses computer simulations, and we want these simulations to run fast on computer hardware, so we make it easy for computer hardware to do. That is why we're using polynomials wherever we can.

That is how you explain polynomials to 8th graders. No taylor series / calculus needed.

If you want to be really fancy you can show the taylor series of the sine and cosine function as a polynomial and how to compute it on a computer. Gives some pretty graphs, is simple and fun.

Just tell them that polynomials can be used to computer sin and cos functions without going into the details of why that works first.

Edit: Just to clarify this: Yes i think that explaining why students should learn stuff is extremely important. In fact i tend to say that the only thing that you really have to do is to motivate the students to learn; then the learning happens by itself.

However note that giving esoteric abstract playful descriptions of things in my opinion does not motivate people to learn stuff. That just makes them go "huh, neat but useless". Giving real world practical examples fulfills exactly the purpose of giving students a reason to learn stuff. Because seeing how one can solve real problems with the tools, one learns to value the tools.