Question

Is `f(x) = 5x^4 - pi(x)^3 + (1/2)` a polynomial function and if so what is the degree?

Answer 1

Yes this is a polynomial function with degree 4 (also called a quartic function).

Explanation:

As is explained here, polynomial functions contain powers which are non-negative integers. The degree is defined as the largest power in the function which does not have a coefficient of zero. You can also tell based on the graph (below) that this is a polynomial. All polynomials have graphs which do not contain breaks, gaps, or other abnormalities. This is a quick way to check if something is a polynomial.
graph{5x^4-pi(x^3)+1/2 [-10, 10, -5, 5]}