Question

What does the degree of a polynomial tell you about its related polynomial function?

Answer 1

the degree indicates many things, but the main information is that if you have a Nth degree prolynomial, you have N roots

Answer 2

Also, if the polynomial has odd degree, there is at least one real root. This helps in the search of other roots, which may be real or complex.
For graphing:

Polynomials of even degree are m-shaped they both start and finish at +\(\infty\) or -\(\infty\), for odd-degrees, they start at +\(\infty\) and finish at -\(\infty\) or vice versa.