$p(x) \in T[x]$. For $\alpha \in T$ to be a multiple root of $p(x)$, a necessary and sufficient condition is that $p(\alpha)=0$ and $p'(\alpha)=0$

Question

Let $T$ be a zone of completeness and $p(x) \in T[x]$. For $\alpha \in T$ to be a multiple root of $p(x)$, a necessary and sufficient condition is that $p(\alpha)=0$ and $p'(\alpha)=0$. Show me. I tried a lot but I don't know how to do it please help.

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score 1 · Answer 1 · answered Jun 23 '21 at 14:14

1

If $\alpha$ is a root of $p(x)$ then $(x-\alpha)$ divides $p(x)$. That is $p(x) = (x-\alpha)q(x)$ for some polynomial $q(x)$. If $\alpha$ is a multiple root then it must also be a root of $q(x)$ (and if $\alpha$ is a root of $q(x)$ then it is a multiple root of $p(x)$).

Then $p'(x) = (x-a)q'(x) + q(x)$. It follows that $p'(\alpha) = 0 \iff q(\alpha) = 0$.

answered Jun 23 '21 at 14:14

Thank you. Are you sure about the solution? – nurylmz Jun 23 '21 at 14:17

$p(x) \in T[x]$. For $\alpha \in T$ to be a multiple root of $p(x)$, a necessary and sufficient condition is that $p(\alpha)=0$ and $p'(\alpha)=0$

1 Answers1