How Many Roots/Factors Can a Polynomial Have?

Stub explanation

A linear equation has either $1$ or $0$ solutions (its graph---a line---intersects the $x$-axis at either $1$ or $0$ points). In other words, a linear polynomial (degree $1$) has at most $1$ root.

A quadratic polynomial (degree $2$) has at most $2$ roots. For instance, this follows from the quadratic formula, but there are other ways to see it as well.

In fact, a degree $n$ polynomial has at most $n$ roots---why?.