Question

if x+y+z=0 and xyz=-1 then prove that 1/(1+x^3)+1/(1+y^3)+ 1/(1+z^3)=1
​

Answer 1

Answer:

Step-by-step explanation:

$1,w,w^{2}$ are roots of x,y,z

So ,

$1+w+w^{2}=0\\\\1*w*w^{2} =-1\\\\w^{3}=-1$

Now , $x=1,y=w,z=w^{2}$

Insert these in the required equation,we get

$\frac{1}{x^{3}}+\frac{1}{y^{3}}+\frac{1}{z^{3}}\\\\ =>\frac{1}{(1)^{3}}+\frac{1}{w^{3} } +\frac{1}{(w^{2})^{3}} \\\\=>1+\frac{1}{-1} +\frac{1}{(w^{3})^{2}}\\\\=>1+-1+\frac{1}{(-1)^{2}} \\\\=>1-1+1\\\\=>1$

Hence proved

Hope helps you