Question

Prove that
$open vertical bar table row cell 3 straight a end cell cell space space minus straight a plus straight b end cell cell space space minus straight a plus straight c end cell row cell negative straight b plus straight a end cell cell 3 straight b end cell cell space minus straight b plus straight c end cell row cell negative straight c plus straight a end cell cell space minus straight c plus straight b end cell cell 3 straight c end cell end table close vertical bar space equals 3 left parenthesis straight a plus straight b plus straight c right parenthesis left parenthesis ab plus bc plus ca right parenthesis$

Solution

Let space space increment space space equals space space open vertical bar table row cell 3 straight a end cell cell space space minus straight a plus straight b end cell cell space space minus straight a plus straight c end cell row cell negative straight b plus straight a end cell cell 3 straight b end cell cell space minus straight b plus straight c end cell row cell negative straight c plus straight a end cell cell space minus straight c plus straight b end cell cell 3 straight c end cell end table close vertical bar space space space space space space space space space space space space space space space equals space open vertical bar table row cell straight a plus straight b plus straight c end cell cell space space minus straight a plus straight b end cell cell space space minus straight a plus straight c end cell row cell straight a plus straight b plus straight c end cell cell 3 straight b end cell cell space minus straight b plus straight c end cell row cell straight a plus straight b plus straight c end cell cell space minus straight c plus straight b end cell cell 3 straight c end cell end table close vertical bar space comma space by space straight C subscript 1 space plus straight C subscript 2 space plus straight C subscript 3 space space space space space space space space space space space space space space equals space space left parenthesis straight a plus straight b plus straight c right parenthesis space space space open vertical bar table row 1 cell space space minus straight a plus straight b end cell cell space space minus straight a plus straight c end cell row 1 cell space 3 straight b end cell cell space minus straight b plus straight c end cell row 1 cell space minus straight c plus straight b end cell cell 3 straight c end cell end table close vertical bar space space space space space space space space space space space space space space space space equals space space left parenthesis straight a plus straight b plus straight c right parenthesis space space open vertical bar table row 1 cell space space minus straight a plus straight b end cell cell space space minus straight a plus straight c end cell row 0 cell space space space straight a plus 2 straight b end cell cell space space space space straight a minus straight b end cell row 0 cell space space straight a minus straight c end cell cell space space space space straight a plus 2 straight c end cell end table close vertical bar space comma space by space straight R subscript 2 minus straight R subscript 1 comma space straight R subscript 3 minus straight R subscript 1 space space space space space space space space space space space space space space equals space left parenthesis straight a plus straight b plus straight c right parenthesis space space space open vertical bar table row cell straight a plus 2 straight b end cell cell space space straight a minus straight b end cell row cell straight a minus straight c end cell cell space space space straight a plus 2 straight c end cell end table close vertical bar space space space space space space space space space space space space space space space equals space space left parenthesis straight a plus straight b plus straight c right parenthesis space space open square brackets table row cell left parenthesis straight a plus 2 straight b right parenthesis left parenthesis straight a plus 2 straight c right parenthesis minus left parenthesis straight a minus straight c right parenthesis left parenthesis straight a minus straight b right parenthesis end cell end table close square brackets space space space space space space space space space space space space space space equals space left parenthesis straight a plus straight b plus straight c right parenthesis space space open square brackets table row cell straight a squared end cell end table plus 2 ca plus 2 ab plus 4 bc minus straight a squared plus ab plus ca minus bc close square brackets space space space space space space space space space space space space space space space space equals space left parenthesis straight a plus straight b plus straight c right parenthesis space space open square brackets 3 ca plus 3 ab plus 3 bc close square brackets space equals 3 left parenthesis straight a plus straight b plus straight c right parenthesis left parenthesis ab plus bc plus ca right parenthesis space space space space space space space space space space space space space space space space space space space

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