TITLE: Repeated roots of a polynomial
QUESTION [5 upvotes]: Whenever a part of the graph of a polynomial is in the form of a parabola whose vertex touches the x axis we conclude that a root is repeated at that point. Why is a root said to be repeated at that point?
How do we conclude it from the shape of the graph?

REPLY [6 votes]: If the graph of a polynomial crosses the $x$-axis at $r$ then $r$ is a root and $x-r$ is a factor. If the graph is tangent to the $x$-axis at $r$ then both the polynomial and its derivative have a root at $r$, and that means $(x-r)^2$ is a factor of the polynomial. That's why we call it a double root.
You can carry this geometric analysis further. For example $0$ is a triple root of the polynomial $x^3$ because the graph there has an inflection point with a horizontal tangent.