---- > [!definition] Definition. ([[root of a polynomial]]) > Let $R$ be a [[ring]] and $f \in R[x]$ a [[polynomial 4|polynomial]]. An element $a \in R$ is called a **root** of $f$ if the evaluation $f(a)=0$. > > Equivalently, $a$ is a root of $f(x) \in R[x]$ if and only if $(x-a)$ [[divides]] $f(x)$. More generally, we say $a$ is a root with **multiplicity** $r$ if $(x-a)^{r}$ [[divides]] $f$ and $(x-a)^{r+1}$ *does not* [[divides|divide]] $f$. ^definition > [!justification] The equivalence. > For one direction:$\begin{align}(x-a) \text{ divides }f & \implies f(x)=g(x)(x-a)\text{ for some } g(x) \in R[x] \\ & \implies f(a)= g(a)(a-a) =0\end{align}$ Conversely, suppose we are given that $f(a)=0$. Let us apply [[the division algorithm for polynomials]] to write $f(x)$ as $f(x)=g(x)(x-a)+r$ (note that $r$ is constant, since $\text{deg }r < \text{deg }(x-a)=1$). Evaluation at $a$ yields $0=r$ and so $(x-a)$ divides $f$, as required. ^justification ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```