Definition of a Polynomial Ring

Let $\mathbb{R}[x]$ be the set of all polynomials in $x$:

$$\mathbb{R}[x]=\left\{a_0,a_{1}x,\dots ,a_n{x}^n:a_n\in \mathbb{R},n\in \mathbb{N}\cup \left\{0\right\}\right\}$$

This is a ring with sum and multiplication of polynomials:

$$(f+g)(x)=f(x)+g(x)\text{and}(f\cdot g)(x)=f(x)\cdot g(x)$$

Note: all polynomial rings are commutative rings, i.e. $a\cdot b=b\cdot a$.