Constants, Powers, Sums, Scalar Multiples and Products of Polynomial Functions

Let $K$ be a field, with additive identity $0$ and multiplicative identity $1$, and let $\mathbb{N}$ be the set of natural numbers. Let $p,q:K\to K$ be polynomial functions on $K$ and let $\lambda\in K$. Write $p+q$, $\lambda p$ and $pq$ for the pointwise sum, scalar multiple and product, given by

$$(p+q)(x)=p(x)+q(x),\qquad (\lambda p)(x)=\lambda\,p(x),\qquad (pq)(x)=p(x)\,q(x)\qquad(x\in K).$$

Then the following hold.

1. (Constants and powers) The map $K\to K$ with constant value $\lambda$ is a polynomial function on $K$; and for every $n\in\mathbb{N}$ the map $K\to K$ sending $x$ to the $n$th power $x^{n}$ is a polynomial function on $K$.

2. (Sums and scalar multiples) $p+q$ and $\lambda p$ are polynomial functions on $K$.

3. (Products) $pq$ is a polynomial function on $K$.