. The equation $x^3 +10 x^2 − 100 x + 1729 = 0$ has at least one complex root $α$ such that $ | α | > 12$.

Question

Show that the equation,

$x^3+10x^2−100x+1729=0$ has at least one complex root $z$ such that $|z|>12$.

Answer 1

Hint:

Note that $1728=12^3$

and if the $x_1,x_2,x_3$ are the three roots with $\text {modulus }\le 12$ Then we have
$$|x_1x_2x_3|\le 12^3$$ $$x_1x_2x_3=(-1)^31729\implies |x_1x_2x_3|=+1729 $$

See Wolfram Alpha

Note: The field of complex numbers includes the field of real numbers as a subfield.