Solve equations using De Moivre's theorem:
1. x^7 + x^4 + x^3 + 1 = 0
2. x^7 - x^4 + x^3 - 1 = 0
I tried multiplying with (x - 1). Also tried putting x^3 = y; didn't work.
Hello, anil86!
I'll walk you through the first one . . .
Solve equations using De Moivre's theorem:
$\displaystyle (1)\;x^7 + x^4 + x^3 + 1 \:=\: 0$
Factor: .$\displaystyle x^4(x^3+1) + (x^3+1) \:=\:0$
Factor: .$\displaystyle (x^3+1)(x^4+1) \:=\:0$
$\displaystyle x^3 + 1 \,=\,0 \quad\Rightarrow\quad x^3 \,=\,-1 \quad\Rightarrow\quad x \,=\,\sqrt[3]{\text{-}1}$
$\displaystyle -1 \:=\:\cos(\pi + 2\pi n) + i\sin(\pi + 2\pi n)$
$\displaystyle (-1)^{\frac{1}{3}} \;=\;\cos\left(\tfrac{\pi}{3} + \tfrac{2\pi}{3}n\right) + i\sin\left(\tfrac{\pi}{3} + \tfrac{2\pi}{3}n\right) \;\text{ for }n = 0,1,2$
$\displaystyle x^4 + 1 \,=\,0 \quad\Rightarrow\quad x^4 \,=\,-1 \quad\Rightarrow\quad x \,=\,\sqrt[4]{\text{-}1}$
$\displaystyle -1 \:=\:\cos(\pi + 2\pi n) + i\sin(\pi + 2\pi n)$
$\displaystyle (-1)^{\frac{1}{4}} \;=\;\cos\left(\tfrac{\pi}{4} + \tfrac{\pi}{2}n\right) + i\sin\left(\tfrac{\pi}{4} + \tfrac{\pi}{2}n\right) \;\text{ for }n = 0,1,2,3$