InterviewSolution
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InterviewSolution
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Prove that the distance of the roots of the equation `|sintheta_1|z^3+|sintheta_2|z^2+|sintheta_3|z+|sintheta_4|=3fromz=0`is greater than `2//3.` |
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Answer» We know that `| sin theta_(k)| lt 1`. Given `|sin theta_(1)|z^(3) + |sin theta_(2)| z^(2) +|sin theta_(3)| z+ |sin theta_(4)| = 3` `or |3|=||sin theta_(1)|z^(3)+|sin theta_(2)| z^(2) +|sin theta_(3)| z+|sin theta_(4)| = 3` `lt 1|z^(3)+z^(2) + z + 1` `lt|z|^(3) +|z|^(2) +|z| +1` `lt 1+|z| +|z|^(2)+ |z|^(3) +|z|^(4) +....oo` `or 3 lt (1)/(1-|z|)` `or 3-3|z| lt 1` `or 2lt 3|z|` `or |z| gt (2)/(3)` |
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