Question

$show space that space open vertical bar table row cell negative straight a left parenthesis straight b squared plus straight c squared minus straight a squared right parenthesis end cell cell 2 straight b cubed end cell cell 2 straight c cubed end cell row cell 2 straight a cubed end cell cell space space space minus straight b left parenthesis straight c squared plus straight a squared plus straight b squared right parenthesis end cell cell 2 straight c cubed end cell row cell 2 straight a cubed end cell cell 2 straight b cubed end cell cell space space space minus straight c left parenthesis straight a squared plus straight b squared space minus straight c squared right parenthesis end cell end table close vertical bar space equals abc space left parenthesis straight a squared plus straight b squared plus straight c squared right parenthesis$

Solution

Let space space increment space space equals space space open vertical bar table row cell negative straight a left parenthesis straight b squared plus straight c squared minus straight a squared right parenthesis end cell cell 2 straight b cubed end cell cell 2 straight c cubed end cell row cell 2 straight a cubed end cell cell space space space minus straight b left parenthesis straight c squared plus straight a squared plus straight b squared right parenthesis end cell cell 2 straight c cubed end cell row cell 2 straight a cubed end cell cell 2 straight b cubed end cell cell space space space minus straight c left parenthesis straight a squared plus straight b squared space minus straight c squared right parenthesis end cell end table close vertical bar space space space space space space space space space space space space space equals space abc space space space space space open vertical bar table row cell negative straight b squared minus straight c squared plus straight a end cell cell 2 straight b squared end cell cell 2 straight c squared end cell row cell 2 straight a squared end cell cell space space space space minus straight c squared minus straight a squared plus straight b squared end cell cell 2 straight c squared end cell row cell 2 straight a squared end cell cell 2 straight b squared end cell cell space space space space space space space space space space space minus straight a squared minus straight b squared plus straight c squared end cell end table close vertical bar space comma space by space taking space straight a comma straight b comma straight c space common space from space straight C subscript 1 comma end subscript straight C subscript 2 comma straight C subscript 3 space space space space space space space space space space space space equals space abc space space space space space space open vertical bar table row cell straight a squared plus straight b squared plus straight c squared end cell cell 2 straight b squared end cell cell 2 straight c squared end cell row cell straight a squared plus straight b squared plus straight c squared end cell cell space space space space space space space space space space minus straight c squared minus straight a squared plus straight b squared end cell cell 2 straight c squared end cell row cell straight a squared plus straight b squared plus straight c squared end cell cell 2 straight b squared end cell cell space space space space space space minus straight a squared minus straight b squared plus straight c squared end cell end table close vertical bar space space comma space by space straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3 space space space space space space space space space space space space equals space abc space left parenthesis straight a squared space plus straight b squared space plus straight c squared space right parenthesis space open vertical bar table row 1 cell 2 straight b squared end cell cell 2 straight c squared end cell row 1 cell negative straight c squared minus straight a squared plus straight b squared end cell cell 2 straight c squared end cell row 1 cell 2 straight b squared end cell cell left parenthesis negative straight a squared minus straight b squared plus straight c 2 right parenthesis end cell end table close vertical bar space space space space space space space space space space space space equals space abc space left parenthesis straight a squared space plus straight b squared space plus straight c squared space right parenthesis space space open square brackets table row cell 1 space open curly brackets negative straight a left parenthesis straight a squared plus straight b squared plus straight c squared close curly brackets. open curly brackets negative left parenthesis straight a squared plus straight b squared plus straight c squared close curly brackets end cell end table close square brackets space space space space space space space space space space space equals space abc space left parenthesis straight a squared space plus straight b squared space plus straight c squared space right parenthesis cubed space

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