system
Given the following post, respond as if you're commenting on Reddit:
human
"Every polynomial of degree n has exactly n roots"

We learned this in class the other day, but what about say f(X)=x²?, There is only one root, but shouldn't there be two according to the prior statement?
gpt
The statement is really more like 'every polynomial of degree n can be factored into the form a(x-c1)(x-c2)...(x-cn)' for constants a,c1,...,cn.' The ci don't have to be distinct.

For convenience, we say that c1,...,cn are the n roots *counted with multiplicity*. That is, in the case if x^(2), it can be said to have two roots: 0, 0.

Why do we do this? Well, nice statements that would be false now become true. For example, consider the following statement:

If p is a monic polynomial of degree n, then the constant term of the polynomial is (-1)^(n) times the product of the roots of p.

(Monic just means the leading x^n coefficient is 1.)

For that statement to be true, you have to count the roots with multiplicity. For example, if p(x) = (x-2)^2 = x^2 - 4x + 4, then the roots of p are 2,2, and indeed the constant term is 4=(-1)^(2)\*2\*2.