How to show that the given equation has at least one complex root ,a s.t |a|>12 How do I show that the equation $x^3+10x^2-100x+1729$$=0$ has at least one complex root $a$ such that $|a|$$>$$12$.

The function necessarily has three complex roots, which we'll call $\alpha$, $\beta$ and $\gamma$. Hence your polynomial can be factored as
$$(x - \alpha) (x - \beta) (x - \gamma) = x^3 + 10x^2 - 100x + 1729$$
Expanding the left hand side, we find that
$$-\alpha \beta \gamma = 1729$$
If it were true that all three roots had moduli less than or equal to $12$, we would have
$$1729 = |\alpha| |\beta| |\gamma| \le 12^3 = 1728$$
a contradiction.
