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Does the equation $x^3+10x^2-100x+1729=0$ fas atleast one complex root $\alpha$ such that $|\alpha|$ $\gt 12$??

Since any cubic equation has atleast one real root, let it be say $k$. Let the other two complex roots be $\alpha+i\beta , \alpha-i\beta$. Then The product of the roots is $(\alpha^2+\beta^2)k=-1729.$ Since $(\alpha^2+\beta^2) \ge 0$, $k\lt0$.

From here I am unable to find any roots or deduce any further.

tattwamasi amrutam
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1 Answers1

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Hint: You are very close. Now suppose $|k| \lt 12$ and $|\alpha + i\beta| \lt 12$ How does that make a problem?

Ross Millikan
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