Question

If the area of the polygon whose vertices are the solutions (in the complex plane) of the equation $x^7+x^6+x^5+x^4+x^3+x^2+x+1=0$, can be expressed in the simplest form as $\frac{a\sqrt{b}+c}{d}$, find the value of $(a+b+c+d)$

Answer 1

Answer: 8

$(x-1)\left(x^7+x^6+\dots \dots ..+x-1\right)=0$
$x=\cos \frac{2m\pi }{8}+i\sin \frac{2m\pi }{8},m=1,2\dots \dots .7$
hence solutionare $e^{i\frac{2\pi }{8}},e^{i\frac{4\pi }{8}},e^{i\frac{6\pi }{8}},e^{i\frac{8\pi }{8}},e^{i\frac{10\pi }{8}},e^{i\frac{12\pi }{8}},e^{i\frac{14\pi }{8}}$
hence polygon $S_1{S}_2\dots \dots S_7$ is as shown

$\text{ Area }=6\times \frac{1}{2}\cdot 1\cdot 1\cdot \frac{1}{\sqrt{2}}+\frac{1}{2}\cdot 1\cdot 1=\frac{3\sqrt{2}}{2}+\frac{1}{2}$
$\text{ Area }=\frac{3\sqrt{2}+1}{2}\equiv \frac{a\sqrt{b}+c}{d}$
$\Rightarrow a+b+c+d=3+2+1+2=8$