Def/Polynomial

 

=Arithmetic operations= Polynomials can be added and multiplied, in such a way that $R[X]$ is a commutative ring whenever $R$ is a commutative ring.

=Evaluation=

If $R$ is a ring, and $P \in R[X]$ is a polynomial, then we can defines::evaluate $P$ at any element $r \in R$. Specifically, if $P = r_0 + r_1 X + \cdots + r_d X^d$ is a polynomial in $R[X]$, the evaluation of $P$ at $r$ is defined by: $$P(r) = r_0 + r_1 r + \cdots + r_d r^d.$$

Evaluation at $r$ is a ring homomorphism from $R[X]$ to $R$.

=Metadata=