1. The problem statement, all variables and given/known data Hello I am starting to learn mathematics, currently working through Fundamentals of University Mathematics (Woodhead Publishing in Mathematics) Colin M. McGregor (Author), John Nimmo (Author), Wilson W. Stothers (Author) http://www.amazon.co.uk/gp/product/0...ls_o01_s00_i00 I have come across a stumbling block with Exercise 1.5. The exercise is to: Show that (x-y)^3 +(y-z)3 + (z-x)^3 = 3(x-y)(y-z)(z-x) That is it no constraints etc. It mentions: "This can be done by expanding out the brackets, but there is a more elegant solution." 2. Relevant equations 3. The attempt at a solution First of all this only seems to hold in special cases as I have substituted random values for x,y and z and they do not agree. It seems more like this should be an inequality (x-y)^3 +(y-z)3 + (z-x)^3 >= 3(x-y)(y-z)(z-x) But that is not the question set. Please note that this is the first chapter and all that has been covered is basic number theory, rational powers, inequalities and divisibility. I am assuming those are the only tools I have at my disposal, I have not been introduced to any identities etc. I decided to expand out the brackets but seem to be stuck attempting that method also (x-y)^3 +(y-z)3 + (z-x)^3 = 3(x-y)(y-z)(z-x) x3-y3-xy(1+2x-2y-y) + y3-z3-yz(1+2y-2z-z) + z3-x3-zx(1+2z-2x-x) = 3(x-y)(y-z)(z-x) all the cubes then cancel -xy(1+2x-2y-y) -yz(1+2y-2z-z) -zx(1+2z-2x-x) = 3(x-y)(y-z)(z-x) Then I start to struggle to see where I can go to make the two equivalent. What I really want to know is this (unmentioned in the solutions section) "elegant solution". The only thing I am aware of is this identity (this is not mentioned in the textbook so I cannot use this to help me) a3+b3+c3-3abc = (a+b+c)(a2+b2+c2-ab-bc-ac) And for this identity if a+b+c=0 then obviously a3+b3+c3-3abc=0 If i substitute (x-y)(y-z)(z-x) for a,b and c respectively a+b+c=x-y+y-z+z-x The x's y's and z's cancel giving a+b+c=z-y+y-z+z-x=0 I think that would seem to be on the right lines but this is not mentioned anywhere in the textbook. Basically I am all muddled up.