Polynomial equations might be written as dot product of factors and powers of x.
For instance, quadratic equation:
$$ ax^2 + bx + c $$
might be rewritten as:
$$ \begin{pmatrix} a & b & c \end{pmatrix} \begin{pmatrix} x^2 \\ x^1 \\ x^0 \end{pmatrix} $$
So to write this even shorter we can put all a, b and c into some kind of vector $\vec{a}$:
$$ a_2 x^2 + a_1 x^1 + a_0 x^0 = \begin{pmatrix} a_2 & a_1 & a_0 \end{pmatrix} \begin{pmatrix} x^2 \\ x^1 \\ x^0 \end{pmatrix} = \vec{a} \begin{pmatrix} x^2 \\ x^1 \\ x^0 \end{pmatrix} $$
But what to do with vector for x? Is there some kind of special notation that represents a vector of powers of x for a given dimension?
Like this:
$$ \vec{a} \begin{pmatrix} x^2 \\ x^1 \\ x^0 \end{pmatrix} = \vec{a} \cdot \vec{x}^2 $$
This way it is possible to write polynomials much shorter, like $\vec{a} \cdot \vec{x}^2$ or $\vec{a} \cdot \vec{x}^5$.
So, do vector-like notations for polynomials exist?