I need to prove the following properties for any two polynomials $p,q$ and $\alpha \in \mathbb{C}$
$$(p + q)(x) = p(x) + q(x)$$ $$(\alpha p)(x) = \alpha(p(x))$$
First of all, these two properties was, for me, the definition of function arithmetic from my pre-calculus days. So my first instinct is that these properties follow directly from the definition of function notation.
Secondly, if I was still going to attempt a proof I think I would do it as follows
proof
let $p,q$ be the following operators
$$ p = a_n\left( \right)^n + \cdots + a_1\left( \right) + a_0$$ $$ q = b_m\left( \right)^m + \cdots + b_1\left( \right) + b_0$$
and now I'll add these operators, but here's where I'm at a loss because I don't know how to add operators. Can I just assume (w.l.o.g) $n > m$ to get
$$p + q = a_n\left( \right)^n + \cdots + (a_m + b_m)\left(\right)^m + \cdots + (a_1+ b_1) \left( \right) + a_0+ b_0$$
These properties seem way too obvious to have to have to be proved I guess. Or is there some nuance I'm missing.
Maybe you can introduce some notation, and then use linear algebra:
If you rewrite $p(x)\ =\ a_n x^n\ +\ a_{n-1} x^{n-1}\ +\ ...+\ a_0$, to be:
$$p = (a_n\ a_{n-1}\ ...\ a_0)$$ $$x = (x^n\ x^{n-1}\ ...\ 1)$$
then you can just make $p(x) := px^{\top}$
Now you can add operators and prove $(p+q)(x) = p(x)+q(x)$ and $\alpha(p(x)) = (\alpha p)(x)$.
As you see it is all matter of how you define things and stay congruent.