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```        If x+ y+ z=0 then prove that (x²+xy+y²)³+ (y²+yz+z²)³+ (z²+zx+x²)³ =
3(x²+xy+y²). (y²+yz+z²)  (z²+zx+x²).
```
7 years ago

105 Points
```
If x+ y+ z=0
Prove that:
(x²+xy+y²)³+ (y²+yz+z²)³+ (z²+zx+x²)³ = 3(x²+xy+y²). (y²+yz+z²) (z²+zx+x²).
Let a = (x²+xy+y²)
b= (y²+yz+z²)
c= (z²+zx+x²)
This equation reduces to proving that a3+b3+c3=3abc
This is possible if:

a+b+c = 0
or a=b=c

We realize by simple substitution(like taking x,y,z = (-1,0,1),(-2,0,2)) that a+b+c is not 0 all the times
Now considering a=b
If and only if
(x²+xy+y²) = (y²+yz+z²)
If and only if
x²+xy = yz+z²
i.e.,        x(x+y)=z(y+z)
Using the fact that x+y+z=0, we see this is nothing but,
x(-z)=z(-x)
Hence a=b=c
=> a3+b3+c3=3abc
which is
(x²+xy+y²)³+ (y²+yz+z²)³+ (z²+zx+x²)³ = 3(x²+xy+y²). (y²+yz+z²) (z²+zx+x²).
Hence proved.
Please feel free to post as many doubts on our discussion forum as  you can. We are all IITians and here to help you in your IIT JEE  preparation.

```
7 years ago
SAGAR SINGH - IIT DELHI
879 Points
```										Dear ajit,
Take LHS
(x²+xy+y²)³+ (y²+yz+z²)³+ (z²+zx+x²)³
Multiply by [(x-y)((y-z)(z-x)]^3 and simplify we will get RHS....

All the best.
Win exciting gifts by answering the questions on Discussion Forum. So help discuss any                      query on askiitians forum and become an Elite Expert     League            askiitian.

Sagar Singh
B.Tech, IIT Delhi

```
7 years ago
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