polynomial function
A polynomial function is a function whose rule is a polynomial — you feed in a number x, and the machine returns the value of the polynomial at that x. For example f(x) = x^2 - 4x + 3 takes an input and hands back an output computed by that expression.
Because polynomials never divide by the variable or take its root, polynomial functions are wonderfully tame: their domain is all real numbers (no value of x is ever forbidden), and their graphs are smooth, unbroken curves with no holes, jumps, or vertical asymptotes.
The degree shapes the graph. A degree-1 function is a straight line, degree 2 is a parabola, degree 3 a cubic with up to two turns, and in general a degree-n function has at most n-1 turning points and at most n real roots (the x-values where the graph crosses the x-axis).
For f(x) = x^2 - 4x + 3, f(1) = 1 - 4 + 3 = 0 and f(3) = 9 - 12 + 3 = 0, so x = 1 and x = 3 are its roots.
Roots are the inputs that make the function equal 0.