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

$(x-1)\left(x^7+x^6+\ldots \ldots . .+x-1\right)=0$
$x =\cos \frac{2 m \pi}{8}+i \sin \frac{2 m \pi}{8}, m =1,2 \ldots \ldots .7$
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 $S _1 S _2 \ldots \ldots S _7$ 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 \quad a+b+c+d=3+2+1+2=8 $