Question

Prove that
$open vertical bar table row cell straight a cubed plus 1 end cell cell space space space straight a squared end cell cell space space straight a end cell row cell straight b cubed plus 1 end cell cell space straight b squared end cell cell space space straight b end cell row cell straight c cubed plus 1 end cell cell straight c squared end cell cell space straight c end cell end table close vertical bar space equals space minus left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight a space straight b space straight c space plus space 1 right parenthesis.$

Solution

Let space increment space equals space open vertical bar table row cell straight a cubed plus 1 end cell cell space space straight a squared end cell cell space space straight a end cell row cell straight b cubed plus 1 end cell cell space space straight b squared end cell cell space straight b end cell row cell straight c cubed plus 1 end cell cell space straight c squared end cell cell space space straight c end cell end table to the power of blank close vertical bar space equals space open vertical bar table row cell straight a cubed end cell cell space space straight a squared end cell cell space straight a end cell row cell straight b cubed end cell cell space straight b squared end cell straight b row cell straight c cubed end cell cell space straight c squared end cell straight c end table close vertical bar space plus space open vertical bar table row 1 cell space space straight a squared end cell cell space space straight a end cell row 1 cell space straight b squared end cell cell space space straight b end cell row 1 cell space straight c squared end cell cell space space straight c end cell end table close vertical bar space space space space space equals negative open vertical bar table row straight a cell space space straight a squared end cell cell space straight a cubed end cell row straight b cell space straight b squared end cell cell space straight b cubed end cell row straight c cell space straight c squared end cell cell space straight c cubed end cell end table close vertical bar minus 1 open vertical bar table row 1 cell space space straight a end cell cell space space straight a squared end cell row 1 cell space straight b end cell cell space space straight b squared end cell row 1 cell space space straight c end cell cell space space straight c squared end cell end table close vertical bar space space space space equals negative abc space open vertical bar table row 1 cell space straight a end cell cell space straight a squared end cell row 1 cell space straight b end cell cell space straight b squared end cell row 1 cell space straight c end cell cell space straight c squared end cell end table close vertical bar minus open vertical bar table row 1 cell space straight a end cell cell space straight a squared end cell row 1 cell space straight b end cell cell space straight b squared end cell row 1 cell space straight c end cell cell space straight c squared end cell end table close vertical bar space equals space minus left parenthesis abc plus 1 right parenthesis space open vertical bar table row 1 cell space straight a end cell cell space straight a squared end cell row 1 cell space straight b end cell cell space straight b squared end cell row 1 cell space straight c end cell cell space straight c squared end cell end table close vertical bar equals negative left parenthesis abc plus 1 right parenthesis space open vertical bar table row 1 cell space space straight a end cell cell space space space straight a squared end cell row 0 cell space space straight b minus straight a end cell cell space space space space straight b squared minus straight a squared end cell row 0 cell space space straight c minus straight a end cell cell space space space straight c squared minus straight a squared end cell end table close vertical bar comma space by space straight R subscript 2 minus straight R subscript 1 comma space straight R subscript 3 minus straight R subscript 1 equals space minus left parenthesis abc plus 1 right parenthesis thin space open vertical bar table row cell straight b minus straight a end cell cell space space space space space straight b squared minus straight a squared end cell row cell straight c minus straight a end cell cell straight c squared minus straight a squared end cell end table close vertical bar equals space minus left parenthesis abc plus 1 right parenthesis thin space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space open vertical bar table row 1 cell space space space space space space straight b plus straight a end cell row 1 cell space space space space space space straight c plus straight a end cell end table close vertical bar equals space minus left parenthesis abc plus 1 right parenthesis thin space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b minus straight a right parenthesis equals negative left parenthesis abc plus 1 right parenthesis thin space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c minus straight b right parenthesis equals negative left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis abc plus 1 right parenthesis therefore space space space space space space open vertical bar table row cell straight a cubed plus 1 end cell cell space space space straight a squared end cell cell space space straight a end cell row cell straight b cubed plus 1 end cell cell space straight b squared end cell cell space straight b end cell row cell straight c cubed plus 1 end cell cell space straight c squared end cell cell space straight c end cell end table close vertical bar space equals space minus left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis abc plus 1 right parenthesis

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