# Is there any geometrical definition of polynomials?

We know that a polynomial in single variable/indeterminate is defined algebraically as-

An expression of the form $a_{n}x^n+a_{n-1}x^{n-1}+\dots+a_{2}x^2+a_{1}x+a_{0}$,

where $a_n,\dots,a_0$ are constants/parameters and $x$ is the indeterminate/variable.

But I was wondering if polynomial have any geometrical meaning.

• In the basis $$(x^0, x^1, x^2, \dots, x^n)$$ the polynomial is represented by $$(a_0, a_1, a_2, \dots, a_n)$$ However, one can change the basis. One could use, say, Legendre polynomials (which are orthogonal). – Rodrigo de Azevedo Sep 16 '16 at 15:34

A curve which after certain successive differentiations gives zero at all points. You will get a straight line parallel to x axis at the end at all points.

The graph of a zero$^{th}$ degree polynomial (a constant) is an horizontal. That of a first degree polynomial as straight line and that of a quadratic is a parabola. The cubic has a center of symmetry which is an inflection point. The higher order polynomials have no particularly remarkable shapes/geometric characteristics.

One can suggest the following properties: if you consider the area under the graph of a polynomial of degree $d$ between the $y$ axis and the vertical at $x$, this area describes a polynomial of degree $d+1$. Conversely, the slope at any point of the graph of a degree $d$ polynomial follows a degree $d-1$ polynomial.

An unusual way to produce the graph of a polynomial:

Take a rectangular sheet of rubberband and draw an horizontal line on it. Then twist the left side to obtain a butterfly shape. (This operation is a transform that multiplies all ordinates by the abscissa, $y\to xy$.)

Next, translate the curve vertically. (This operation adds a constant to the ordinates, $y\to y+c$.)

By repeating this pair of operations $d$ times, the line turns to the graph of a degree $d$ polynomial.

You should have a look at any introduction to algebric geometry. The basic objects of algebric geometry are algebric varieties, i.e. the sets of zero of one or several given polynomials. If you want a basic example, the unit circle in $\mathbb{R}^2$ is an algebric variety, because it is the set where the polynomial $x^2+y^2-1$ vanishes.

However, the polynomials considered in algebric geometry have several variables. So maybe it does not answer properly your question...