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Lower bound of roots (TIFR 2013 problem 4)

Question:

True/False:

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

Hint: What happens if \(|\alpha| \le 12\) for all roots \(\alpha?\)

Discussion: Let \(\alpha_1\),\(\alpha_2\),\(\alpha_3\) be roots of the equation. (These are complex numbers and existence is guranteed by the Fundamental Theorem of Algebra). What do we already know about these roots? We know the sum of roots, sum of product of roots taken two at a time, and product of roots. These are expressible by the coefficients of the equation.

Suppose \(|\alpha_1|\le12\),\(|\alpha_2|\le12\),\(|\alpha_3|\le12\).

Then \(|\alpha_1\alpha_2\alpha_3|=|\alpha_1||\alpha_2||\alpha_3|\le12^3=1728\)

But, the product of roots is \(-1729\). That is,  \(|\alpha_1\alpha_2\alpha_3|=1729\)

Therefore things go wrong if we suppose \(|\alpha| \le 12\) for all roots \(\alpha\). Therefore the negation of this statement must be true. What is the negation? Of course, \(\le\) is replaced by \(\gt\) and for all is replaced by for some.

Hence the statement The equation \(x^3 + 10x^2 − 100x + 1729 = 0\) has at least one complex root α such that \(|α| > 12\) is True.

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