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I stumbled upon the question that 'Factorize the polynomial $$p(x)=x^4+x^3+(1+i)x^2+(1-i)x+3i$$

It is commonly known that $\mathbb C$ is algebraically closed. So, any polynomial has at least one complex root. Obviously, there are at most $4$ roots of $p(x)$. I want to learn is there a way to find possible roots of $p(x)$ so as to write $p(x)$ in terms of linear factors without using calculators.

In other words, how one can factorize any polynomial in $\mathbb C$? without knowing any root of it. Thanks in advance.

Fuat Ray
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  • quartics can be solved by radicals, but there's no guarantee of getting pleasant solutions. This one looks pretty typical – lulu May 29 '24 at 17:56
  • Higher degree polynomials must, in general, be solved by numerical means. Frankly, I'd solve generic cubics and quartics numerically as well. These horrific expressions in radicals tend to be less than useful. – lulu May 29 '24 at 18:02
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    Just in case the question was about factorization over $\Bbb{Q}(i)$. This is irreducible! Modulo the prime $1+i$ it is equivalent to $\overline{p}=x^4+x^3+1$ which is one of the three irreducible quartics modulo $2$. And $\Bbb{Z}[i]/(1+i)\simeq \Bbb{Z}/(2)$. – Jyrki Lahtonen May 29 '24 at 18:08

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