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.
 A: 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.
A: 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.
A: 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...
