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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). Answer: 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. Win exciting gifts by answering the questions on Discussion Forum..
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).
Answer:
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:
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)
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
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.
Win exciting gifts by answering the questions on Discussion Forum..
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....
Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation. 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. Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar respectively : Click here to download the toolbar.. Askiitians Expert Sagar Singh B.Tech, IIT Delhi
Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation.
All the best.
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B.Tech, IIT Delhi
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