Final answer:
The statement that a third-degree polynomial function with integer coefficients can have no real zeros is false. Such a polynomial must have at least one real zero due to the Fundamental Theorem of Algebra and graphical behavior of cubic functions.
Step-by-step explanation:
The statement given is: "It is possible for a third-degree polynomial function with integer coefficients to have no real zeros." This statement is false. According to the Fundamental Theorem of Algebra, every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. A third-degree polynomial must have three roots in the complex number system which includes real numbers and imaginary numbers. Since the polynomial has integer coefficients, if it has any complex roots, they must occur in conjugate pairs. This means, for a third-degree polynomial, there must be at least one real zero since it's not possible to form three roots as pairs.
Moreover, by considering Two-Dimensional (x-y) Graphing, a third-degree polynomial, which is a cubic function, will always cross the x-axis at least once, indicating a real zero. The function can have one or three real zeroes, but it cannot have none.