MY CART (5)

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

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: R

There are no items in this cart.
Continue Shopping
Menu
Ajit Siddharth Grade:
        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).
6 years ago

Answers : (2)

suryakanth AskiitiansExpert-IITB
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).


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..



6 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....



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


6 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies
  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: R 15,000
  • View Details

Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!! Click Here for details