Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Please check your email for login details.
MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping

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).

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).

Grade:

2 Answers

suryakanth AskiitiansExpert-IITB
105 Points
10 years ago

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..

SAGAR SINGH - IIT DELHI
879 Points
10 years ago

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

Think You Can Provide A Better Answer ?

Provide a better Answer & Earn Cool Goodies See our forum point policy

ASK QUESTION

Get your questions answered by the expert for free