Repeated roots of a polynomial

Question

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?

Answer 1

It is repeated, graphically speaking, because if you move the curve downwards by a small amount, the single (double) root splits into two (or more: this depends on the function) distinct points.

You can see it on the graph by the fact the $x$-axis is tangent to the graph at that point.

Answer 2

The equation $$ (x-10)(x-11)(x-85) = 0 $$ Has three roots: $10$, $11$, and $85$.

Of the equation $$ (x-10)(x-10)(x-85) $$ it can be said that it has three roots but two of them are equal to each other.

The solution to $ax^2+bx+c=0$, when $a\ne0$, is $x= \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$.

This formula gives TWO solutions, but if $b^2-4ac=0$, then both solutions are the same number.

Answer 3

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.