User:IssaRice/Degree of polynomial

Why is the definition of degree of polynomial what it is?

First attempt: a polynomial like $a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}$ should have degree $n$. This works for a polynomial like $4x^{2}-5x+8$, which has degree 2 according to the definition, as we expect. But now what if we had $0x^{3}+4x^{2}-5x+8$? This is the same polynomial, but it now has degree 3 according to the definition. So actually degrees aren't unique, which is a problem. We want to rule out cases where the leading coefficient is zero.

Second attempt: a polynomial like $a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}$, where $a_{n}\neq 0$, has degree $n$. Now $4x^{2}-5x+8$ still has degree 2, and $0x^{3}+4x^{2}-5x+8$ (the same polynomial) does not have degree 3, since the leading coefficient is zero. We're still not sure if the degree is unique (after all, maybe the polynomial can be written in some crazy way where it has degree 3), but there isn't any obvious problem now. What about $0$, the zero polynomial? this has leading coefficient zero, so it's actually not clear what the degree is -- we've left this case undefined. All the other constant polynomials have degree 0, so it might seem natural to assign a degree of 0. but actually, if we look at products of polynomials it will seem more natural to assign a degree of $-\infty$.