Does every quadratic polynomial defined in real numbers always define a function? 
Does every quadratic (or any degree) polynomial defined in real numbers always define a function?

For example, let $a,b,c\in\mathbb R$ and define
$$P(x)=ax^2+bx+c$$
Can we define this polynomial as a function?
$$f:\mathbb R\rightarrow \mathbb 
R $$ via $$f(x)=ax^2+bx+c$$
I can not see any mistake here, but I'm afraid to use definitions sloppy. That's why I want to ask here.
 A: I think @peter.petrov gave a nice answer to your question already.
To answer your question in the comments concerning Polynomials on matrices:
A Polynomial $P$ can always be defined if you have multiplication and addition in your structure, i.e. matrices can be multiplied and added together. Take for example
$$P(x):=2x^2+x$$ where $P: \mathbb{R}^{2 \times 2}\to\mathbb{R}^{2 \times 2}$. Then, you can choose any $x\in\mathbb{R}^{2\times 2}$ to evaluate $P$ at -  for example, choose $x=\begin{pmatrix}1&2\\0&1\end{pmatrix}$. Then
\begin{align}
P\left( \begin{pmatrix}1&2 \\ 0&1\end{pmatrix} \right)&=2 \begin{pmatrix}1&2 \\ 0&1\end{pmatrix}^2+\begin{pmatrix}1&2 \\ 0&1\end{pmatrix}
\\&=\begin{pmatrix}1&4\\0 &1\end{pmatrix}+\begin{pmatrix}1&2 \\0&1 \end{pmatrix}\\
&=\begin{pmatrix}2&6 \\ 0&2\end{pmatrix}
\end{align}
As you can see, a Polynomial is just a function that looks a specific way (variable with exponent multiplied with coefficient and added together) and can be evaluated at a specific $x$. Hope this cleared things up!
