Simple definition of polynomial.

A polynomial is an expression of the form $a_n{x}^n+a_{(n-1)}{x}^{(n-1)}+a_{(n-2)}{x}^{(n-2)}+........+a_{1}x+a_0$,
Where, all $a_i$'s belongs to set of real numbers, $n$ is set of all non negative integers and $x$ is called variable or indeterminant.
In other words, an expression contains variables and coefficients, that involves the operation of addition, subtraction, multiplication(except division) and non-negative integer exponentiation of variables is called polynomial.
Example: $x^2-4x+1$ is a polynomial.