Which method is primarily used for analyzing the behavior of a polynomial?

The method of finding the degrees and coefficients of a polynomial is fundamental in analyzing its behavior. The degree of the polynomial indicates the highest power of the variable, which in turn determines the polynomial's end behavior and the number of possible turning points. For example, a polynomial of odd degree will have ends that go in opposite directions, while an even degree polynomial will have ends that go in the same direction.

Additionally, the coefficients play a critical role in shaping the graph of the polynomial. They influence the steepness, direction, and overall appearance of the graph. By understanding both the degree and the coefficients, one can make predictions about the polynomial’s behavior without needing to fully graph it. This approach lays the groundwork for further analysis, including factoring and identifying roots, ultimately leading to a complete understanding of the polynomial's characteristics.

In contrast, while the other methods like graph symmetry, factoring, and understanding discontinuities are helpful for specific analyses or scenarios, they do not provide as foundational a perspective on the polynomial's overall behavior as the degree and coefficients do.