(x+y)3  + ( y + z)3  + ( z+x)3 - 3(x+y)(y+z)(z+x) = 2 ( x3 + y3 + z3 - 3xyz)

Asked by Vikas | 23rd Jan, 2018, 11:07: PM

Expert Answer:

begin mathsize 16px style Let space straight x plus straight y equals straight a comma space straight y plus straight z equals straight b space and space straight z plus straight x equals straight c
Then comma
left parenthesis straight x plus straight y right parenthesis cubed plus left parenthesis straight y plus straight z right parenthesis cubed plus left parenthesis straight z plus straight x right parenthesis cubed minus 3 left parenthesis straight x plus straight y right parenthesis left parenthesis straight y plus straight z right parenthesis left parenthesis straight z plus straight x right parenthesis
equals straight a cubed plus straight b cubed plus straight c cubed minus 3 abc
equals left parenthesis straight a plus straight b plus straight c right parenthesis left parenthesis straight a squared plus straight b squared plus straight c squared minus ab minus bc minus ca right parenthesis
equals left square bracket left parenthesis straight x plus straight y right parenthesis plus left parenthesis straight y plus straight z right parenthesis plus left parenthesis straight z plus straight x right parenthesis right square bracket left square bracket left parenthesis straight x plus straight y right parenthesis squared plus left parenthesis straight y plus straight z right parenthesis squared plus left parenthesis straight z plus straight x right parenthesis squared minus left parenthesis straight x plus straight y right parenthesis left parenthesis straight y plus straight z right parenthesis minus left parenthesis straight y plus straight z right parenthesis left parenthesis straight z plus straight x right parenthesis minus left parenthesis straight z plus straight x right parenthesis left parenthesis straight x plus straight y right parenthesis right square bracket
equals 2 left parenthesis straight x plus straight y plus straight z right parenthesis left square bracket straight x squared plus straight y squared plus 2 xy plus straight y squared plus straight z squared plus 2 yz plus straight z squared plus straight x squared plus 2 xz minus left parenthesis xy plus xz plus straight y squared plus yz right parenthesis
space space space space minus left parenthesis yz plus xy plus straight z squared plus xz right parenthesis minus left parenthesis xz plus yz plus straight x squared plus xy right parenthesis right square bracket
equals 2 left parenthesis straight x plus straight y plus straight z right parenthesis left parenthesis straight x squared plus straight y squared plus 2 xy plus straight y squared plus straight z squared plus 2 yz plus straight z squared plus straight x squared plus 2 xz minus xy minus xz minus straight y squared minus yz minus yz minus xy minus straight z squared minus xz minus xz minus yz minus straight x squared minus xy right parenthesis
equals 2 left parenthesis straight x plus straight y plus straight z right parenthesis left parenthesis straight x squared plus straight y squared plus straight z squared minus xy minus yz minus zx right parenthesis
equals 2 left parenthesis straight x cubed plus straight y cubed plus straight z cubed minus 3 xyz right parenthesis end style

Answered by Rashmi Khot | 24th Jan, 2018, 09:41: AM