Question

Solve the following equations in complex numbers and write your answer in polar and rectangular form
z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z = 0.

Answer 1

$\textbf{Given:}$

$z^7+z^6+z^5+z^4+z^3+z^2+z=0$

$\textbf{To find:}$

$\text{Roots of the equation}$

$\textbf{Solution:}$

$\text{Consider,}$

$z^7+z^6+z^5+z^4+z^3+z^2+z=0$

$\text{This is a finite G.P series containing 7 terms}$

$\text{Using the formula \bf\,S_n=\dfrac{a(^n-1)}{r-1} , we get}$

$\dfrac{z(z^7-1)}{z-1}=0$

$\implies\,z(z^7-1)=0$

$\implies\,z=0\;\text{or}\;z^7-1=0$

$z^7-1=0$

$z^7=1$

$z=1^{\frac{1}{7}}$

$z=[cos0+i\,sin0]^{\frac{1}{7}}$

$z=[cos\,2k\pi+i\,sin\,2k\pi]^{\frac{1}{7}}$

$\text{Using demovire's theorem, we get}$

$\bf\,z=cos\,\frac{2k\pi}{7}+i\,sin\frac{2k\pi}{7}$   $\text{k=0,1,2...6}$

$\therefore\textbf{The required roots are}$

$\bf\,z=0$

$\bf\,z=cos\,0+i\,sin0$

$\bf\,z=cos\,\frac{2\pi}{7}+i\,sin\frac{2\pi}{7}$

$\bf\,z=cos\,\frac{4\pi}{7}+i\,sin\frac{4\pi}{7}$

$\bf\,z=cos\,\frac{6\pi}{7}+i\,sin\frac{6\pi}{7}$

$\bf\,z=cos\,\frac{8\pi}{7}+i\,sin\frac{8\pi}{7}$

$\bf\,z=cos\,\frac{10\pi}{7}+i\,sin\frac{10\pi}{7}$

$\bf\,z=cos\,\frac{12\pi}{7}+i\,sin\frac{12\pi}{7}$