Is it appropriate to use iff in this case? I don't know if there is another case which is sufficient to show that $x$ is a root of a polynomial which would mean it is NOT iff.
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1"I don't know if there is another case" $;-;$ What is the definition for the "root of a polynomial" you are using? – dxiv Mar 07 '22 at 07:21
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Maybe you mean this? – Miguel Mar 07 '22 at 07:33
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I assume that by a root of a polynomial $p$ you mean a number $a$ such that $p(x)= q(x)(x-a)$ for some polynomial $q$. By polynomial division, you can always write $p(x)=(x-a)q(x)+b$, where $b$ is a scalar. Then $p(a)=b$, so desired factorization exists if and only if $p(a)=0$.
Blazej
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Does this also hold true if it is a polynomial with complex coefficients? – user71207 Mar 08 '22 at 00:08
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Yes, you can perform polynomial division also for complex polynomials (try proving this!), so the same argument will work. – Blazej Mar 08 '22 at 18:41