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Question Number 123824 by Snail last updated on 28/Nov/20
Show that for all real numbers (x/y/z) satisfying x+y+z=0 and xy +yz+zx=−3 the value of expression x^3 y+y^3 z +z^3 x is a constant
Showthatforallrealnumbers(x/y/z)satisfyingx+y+z=0andxy+yz+zx=−3thevalueofexpressionx3y+y3z+z3xisaconstant
Commented by Snail last updated on 28/Nov/20
I think u have done a mistake u have written x^3 y+y^3 z+z^3 x=A(let) and xy^3 +yz^3 +x^3 z=B(let) thenu have done A=−18−B⇒(a) and then B=−18−A But that does not imply A=B
Ithinkuhavedoneamistakeuhavewrittenx3y+y3z+z3x=A(let)andxy3+yz3+x3z=B(let)thenuhavedoneA=−18−B⇒(a)andthenB=−18−AButthatdoesnotimplyA=B
Answered by MJS_new last updated on 28/Nov/20
z=−(x+y) ⇒ xy+yz+zx=−(x^2 +xy+y^2 )∧x^3 y+y^3 z+z^3 y=−(x^2 +xy+y^2 )^2 =−9
z=−(x+y)⇒xy+yz+zx=−(x2+xy+y2)∧x3y+y3z+z3y=−(x2+xy+y2)2=−9
Commented by Dwaipayan Shikari last updated on 29/Nov/20
z=−(x+y) xy+yz+xz=xy+y(−x−y)−(x+y)x=−(x^2 +xy+y^2 ) x^3 y+y^3 z+z^3 y =x^3 y−y^3 (x+y)−(x+y)^3 y =−(x^2 +xy+y^2 )^2 =−9
z=−(x+y)xy+yz+xz=xy+y(−x−y)−(x+y)x=−(x2+xy+y2)x3y+y3z+z3y=x3y−y3(x+y)−(x+y)3y=−(x2+xy+y2)2=−9
Commented by Snail last updated on 29/Nov/20
I can′t understand the process
Ican′tunderstandtheprocess

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