Menu Close

x-3-1-x-1-x-2-x-1-x-1-and-x-1-3-i-2-w-1-2-3-i-2-and-w-2-1-2-3-2-i-w-3-1-similarly-x-3-1-0-x-1-and-x-w-2-and-x-w-




Question Number 191640 by vishal1234 last updated on 28/Apr/23
x^3 −1 = (x−1) (x^2 +x+1)  ⇒ x = 1 and x = ((−1±(√3) i)/2)  ⇒ w = −(1/2) + (((√3)i )/2) and w^2  = −(1/2) − ((√3)/2) i  ⇒ w^3  = 1  similarly x^3  + 1= 0  ⇒ x = −1 and x =−w^2  and x = −w
$${x}^{\mathrm{3}} −\mathrm{1}\:=\:\left({x}−\mathrm{1}\right)\:\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right) \\ $$$$\Rightarrow\:{x}\:=\:\mathrm{1}\:{and}\:{x}\:=\:\frac{−\mathrm{1}\pm\sqrt{\mathrm{3}}\:{i}}{\mathrm{2}} \\ $$$$\Rightarrow\:{w}\:=\:−\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\sqrt{\mathrm{3}}{i}\:}{\mathrm{2}}\:{and}\:{w}^{\mathrm{2}} \:=\:−\frac{\mathrm{1}}{\mathrm{2}}\:−\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:{i} \\ $$$$\Rightarrow\:{w}^{\mathrm{3}} \:=\:\mathrm{1} \\ $$$${similarly}\:{x}^{\mathrm{3}} \:+\:\mathrm{1}=\:\mathrm{0} \\ $$$$\Rightarrow\:{x}\:=\:−\mathrm{1}\:{and}\:{x}\:=−{w}^{\mathrm{2}} \:{and}\:{x}\:=\:−{w} \\ $$
Commented by a.lgnaoui last updated on 28/Apr/23
w=−(1/2)+((√3)/2)i=j      (j=e^(i((2𝛑)/3)) )
$$\mathrm{w}=−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i}=\boldsymbol{\mathrm{j}}\:\:\:\:\:\:\left(\boldsymbol{\mathrm{j}}=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{i}}\frac{\mathrm{2}\boldsymbol{\pi}}{\mathrm{3}}} \right) \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *