Question

$prove space that open vertical bar table row cell 1 plus straight a squared plus straight b squared end cell cell 2 ab end cell cell negative 2 straight b end cell row cell 2 ab end cell cell 1 minus straight a squared space plus straight b squared end cell cell 2 straight a end cell row cell 2 straight b end cell cell negative 2 straight a end cell cell 1 minus straight a squared minus straight b squared end cell end table close vertical bar space equals left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis cubed$

Solution

Let space space increment space equals space open vertical bar table row cell 1 plus straight a squared plus straight b squared end cell cell 2 ab end cell cell negative 2 straight b end cell row cell 2 ab end cell cell 1 minus straight a squared space plus straight b squared end cell cell 2 straight a end cell row cell 2 straight b end cell cell negative 2 straight a end cell cell 1 minus straight a squared minus straight b squared end cell end table close vertical bar space space space space space space space space space space space space space space equals space open vertical bar table row cell 1 plus straight a squared plus straight b squared end cell cell 2 ab end cell cell negative 2 straight b end cell row 0 cell 1 minus straight a squared space plus straight b squared end cell cell 2 straight a end cell row cell straight b space left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis space end cell cell negative straight a left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis end cell cell 1 minus straight a squared minus straight b squared end cell end table close vertical bar space by space straight C subscript 1 space minus straight b space straight C subscript 3 space comma space straight C subscript 2 space plus straight a space straight C subscript 3 space space space space space space space space space space space space equals space left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis space squared space space space open vertical bar table row 1 cell space space space space space space space space 0 end cell cell negative 2 straight b end cell row 0 cell space space space space space space space 1 end cell cell 2 straight a end cell row cell straight b space end cell cell space space space space minus straight a end cell cell space space space space space 1 minus straight a squared minus straight b squared end cell end table close vertical bar space by space taking space 1 plus straight a squared space plus straight b squared space common space from space straight C subscript 1 space as space well space as space space straight C subscript 2 space space space space space space space space space space space space equals space left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis space squared space space space open vertical bar table row 1 cell space space space space space space space space 0 end cell cell negative 2 straight b end cell row 0 cell space space space space space space space 1 end cell cell 2 straight a end cell row cell straight b space end cell cell space space space space minus straight a end cell cell space space space space space 1 minus straight a squared minus straight b squared end cell end table close vertical bar space comma space by space space straight C subscript 3 space plus 2 straight b. straight C subscript 1 space space space space space space space space space space space equals space left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis space squared space space space open vertical bar table row 1 cell 2 straight a end cell row cell negative straight a end cell cell space space space space space 1 minus straight a squared plus straight b squared end cell end table close vertical bar space comma space expanding space by space space straight R subscript 1 space space space space space space space space space space space equals space left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis space squared space space open square brackets table row cell 1 minus straight a squared plus straight b squared end cell end table plus 2 straight a squared close square brackets space equals space left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis space squared space space left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis space squared space space space space space space space space space space space space equals space space left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis space cubed space space space space space space space space space space space space space space space space

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