An algebra problem by Keshav Tiwari

Algebra Level 5

Let $z = x + iy$ for real numbers $x$ and $y$ , and $x \ne -\dfrac12$ .

Find the number of values of $z$ satisfying $|z|^{n} = z^{2} |z|^{n-2} +z|z|^{n-2} +1$ for natural number $n> 1$ .

Clarification: $i=\sqrt{-1}$ .

1 3 2 0

1 solution

Mark Hennings
Nov 8, 2016

We have $(z + z^2)|z|^{n-2} \; = \; |z|^n - 1$ and hence $z + z^2$ is real. Thus the imaginary part of $z + z^2$ is equal to $0$ , and hence $y + 2xy \; = \; (1 + 2x)y \; = \; 0$ We are told that $x \neq -\tfrac12$ , so it follows that $y=0$ , and hence $z$ is real. Thus $|z|^n + z|z|^{n-2} \; = \; (z^2+z)|z|^{n-2} \; = \; |z|^n - 1$ and hence $z|z|^{n-2} = -1$ , which implies that $z=-1$ . There is only one possible value of $z$ .

An algebra problem by Keshav Tiwari

1 solution

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