Question

What is the relationship between the “root”, an “x-intercept”, and the “solution” of a polynomial function?

Answer 1

X-intercepts need to be real numbers; the others can be complex.

Answer 2

The terms used depend on what you're talking about.
e.g. f(x)=x^2+3x+2
The "zeroes" of the function, f(x), are -1 and -2.
The "roots" of the polynomial, x^2+3x+2 are -1 and -2
The "solutions of the equation, x^2+3x+2=0 are x = -1 and x = -2

Answer 3

What geoffb said is also correct. You speak of x-intercepts when talking about the graph of the function. If a zero of the function isn't a real number, then it won't show up on the graph, but it is still a solution to the equation.