Question

show that
$open vertical bar table row cell straight x minus straight y minus straight z end cell cell 2 straight x end cell cell 2 straight x end cell row cell 2 straight y end cell cell space space space straight y minus straight z minus straight x end cell cell 3 straight x end cell row cell 2 straight z end cell cell 2 space straight z end cell cell space space straight z minus straight x minus straight y end cell end table close vertical bar space equals space left parenthesis x plus y plus z right parenthesis cubed$

Solution

Let space space space increment space equals space open vertical bar table row cell straight x minus straight y minus straight z end cell cell 2 straight x end cell cell 2 straight x end cell row cell 2 straight y end cell cell space space space straight y minus straight z minus straight x end cell cell 3 straight x end cell row cell 2 straight z end cell cell 2 space straight z end cell cell space space straight z minus straight x minus straight y end cell end table close vertical bar space space space space space space space space space space space space space space equals space space space open vertical bar table row cell straight x minus straight y minus straight z end cell cell 2 straight x end cell cell 2 straight x end cell row cell 2 straight y end cell cell space space space straight y minus straight z minus straight x end cell cell 3 straight x end cell row cell 2 straight z end cell cell 2 space straight z end cell cell space space straight z minus straight x minus straight y end cell end table close vertical bar space comma space by space straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3 Taking space out space straight x plus straight y plus straight z space common space from space straight R subscript 1 equals space left parenthesis straight x plus straight y plus straight z right parenthesis space space space open vertical bar table row cell straight x minus straight y minus straight z end cell cell 2 straight x end cell cell 2 straight x end cell row cell 2 straight y end cell cell space space space straight y minus straight z minus straight x end cell cell 3 straight x end cell row cell 2 straight z end cell cell 2 space straight z end cell cell space space straight z minus straight x minus straight y end cell end table close vertical bar equals space left parenthesis straight x plus straight y plus straight z right parenthesis space space space open vertical bar table row 1 0 0 row cell 2 straight y end cell cell space space space space minus straight x minus straight y minus straight z end cell 0 row cell 2 straight z end cell 0 cell space minus straight x minus straight y minus straight z end cell end table close vertical bar space by space straight C subscript 2 minus straight C subscript 1 comma space straight C subscript 3 minus straight C subscript 1 equals left parenthesis straight x plus straight y plus straight z right parenthesis space space space space open vertical bar table row cell negative straight x minus straight y minus straight z end cell 0 row 0 cell negative straight x minus straight y minus straight z end cell end table close vertical bar space comma space expanding space by space first space row equals left parenthesis straight x plus straight y plus straight z right parenthesis space open square brackets table row cell left parenthesis straight x minus straight y minus straight z right parenthesis left parenthesis negative straight x minus straight y minus straight z right parenthesis minus 0.0 end cell end table close square brackets equals space left parenthesis straight x plus straight y plus straight z right parenthesis space left parenthesis straight x plus straight y plus straight z right parenthesis space left parenthesis straight x plus straight y plus straight z right parenthesis equals space left parenthesis straight x plus straight y plus straight z right parenthesis cubed space space

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