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Let $A \in \mathbb{C}$ be a nonzero complex number.

What is the solution of $x_1x_2x_3x_4x_5 + A = 0 $ in the field of complex numbers $\mathbb{C}$ ? Or more generally the solution of $x_1\cdots x_n + A = 0$ in $\mathbb{C}$ ?

Thanks for your attention guys :)

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Are the $X_i$'s complex numbers too? In that case, you can choose any non-zero $X_1, ... X_4$ and then solve for $X_5$. Is that what you meant? –  Ted Sep 4 '11 at 4:09
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You can let $X_1,\ldots,X_{n-1}$ be any nonzero complex numbers, and choose $X_n = -A/(X_1\cdots X_{n-1})$. –  Arturo Magidin Sep 4 '11 at 4:09
    
@Ted Please consider converting your comment into an answer, so that this question gets removed from the unanswered tab. If you do so, it is helpful to post it to this chat room to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see here, here or here. –  Julian Kuelshammer Jun 10 '13 at 8:44
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1 Answer

This CW post intends to remove the question from the unanswered queue.


As remarked in the comments the general solution of $X_1\cdots X_n+A=0$ is $$\{(X_1,\dots, X_n)|X_1\neq 0,\dots, X_{n-1}\neq 0, X_n=-\frac{A}{X_1\cdots X_{n-1}}\}$$ This follows by observing that if one of the $X_i$ is zero, then because of $A$ being non-zero, this will never be a solution and in the case of all of them being non-zero solving for $X_n$.

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