if x+y+z=0, prove that

Asked by sourabh7jain | 23rd Jun, 2016, 12:23: PM

Expert Answer:

LHS space equals open vertical bar table row xa yb zc row yc za xb row zb xc ya end table close vertical bar
equals xa left parenthesis yza squared minus straight x squared bc right parenthesis minus yb left parenthesis acy squared minus zxb squared right parenthesis plus zc left parenthesis yxc squared minus straight z squared ab right parenthesis
equals xyz left parenthesis straight a cubed plus straight b cubed plus straight c cubed right parenthesis minus abc left parenthesis straight x cubed plus straight y cubed plus straight z cubed right parenthesis
equals xyz left parenthesis straight a cubed plus straight b cubed plus straight c cubed right parenthesis minus abc left parenthesis xyz right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because straight x plus straight y plus straight z space equals space 0 close square brackets
equals xyz 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


RHS space equals xyz space open vertical bar table row straight a straight b straight c row straight c straight a straight b row straight b straight c straight a end table close vertical bar
equals xyz open vertical bar table row cell straight a plus straight b plus straight c end cell cell straight a plus straight b plus straight c end cell cell straight a plus straight b plus straight c end cell row straight c straight a straight b row straight b straight c straight a end table close vertical bar space space space space space space space space space space open square brackets straight R subscript 1 rightwards arrow straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3 close square brackets
equals xyz left parenthesis straight a plus straight b plus straight c right parenthesis open vertical bar table row 1 1 1 row straight c straight a straight b row straight b straight c straight a end table close vertical bar
equals xyz left parenthesis straight a plus straight b plus straight c right parenthesis open square brackets straight a squared minus bc plus ac plus straight b squared plus straight c squared minus ab close square brackets
equals xyz left parenthesis straight a plus straight b plus straight c right parenthesis open square brackets straight a squared plus straight b squared plus straight c squared minus ab minus bc minus ca close square brackets

LHS equals space RHS
Proved

Answered by Vijaykumar Wani | 23rd Jun, 2016, 05:44: PM

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