Polynomial ring/Field/Zero/Linear factor/Fact

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. Let ${\displaystyle {}P\in K[X]}$ be a polynomial and ${\displaystyle {}a\in K}$.

Then ${\displaystyle {}a}$ is a

zero

of ${\displaystyle {}P}$ if and only if ${\displaystyle {}P}$ is a multiple of the linear polynomial ${\displaystyle {}X-a}$.