Prove the following identities:
$open vertical bar table row 1 cell space space space straight x end cell cell space space space straight x cubed end cell row 1 cell space space straight y end cell cell space space straight y cubed end cell row 1 cell space straight z end cell cell space space straight z cubed end cell end table close vertical bar space equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight x plus straight y plus straight z right parenthesis$

Solution

Let $increment space equals space open vertical bar table row 1 cell space space space space straight x end cell cell space space space straight x cubed end cell row 1 cell space space space straight y end cell cell space space space straight y cubed end cell row 1 cell space space straight z end cell cell space space space straight z cubed end cell end table close vertical bar$
$equals space open vertical bar table row 1 cell space space space space space space straight x end cell cell space space space space straight x cubed end cell row 0 cell space space space space space straight y minus straight x end cell cell space space space space space space straight y cubed minus straight x cubed end cell row 0 cell space space space straight z minus straight x end cell cell space space space straight z cubed minus straight x cubed end cell end table close vertical bar comma space by space straight R subscript 2 minus straight R subscript 1 comma space space straight R subscript 3 minus straight R subscript 1 equals space open vertical bar table row cell straight y minus straight x end cell cell space space space space space straight y cubed minus straight x cubed end cell row cell straight z minus straight x end cell cell space space space space space space straight z cubed minus straight x cubed end cell end table close vertical bar space equals space open vertical bar table row cell straight y minus straight x end cell cell space space space space space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight y squared plus straight x squared plus yx right parenthesis end cell row cell straight z minus straight x end cell cell space space space space space space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z squared plus straight x squared plus zx right parenthesis end cell end table close vertical bar equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space space open vertical bar table row 1 cell space space space space space straight y squared plus straight x squared plus yx end cell row 1 cell space space space space straight z squared plus straight x squared plus zx end cell end table close vertical bar space equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left square bracket straight z squared plus straight x squared plus zx minus straight y squared minus straight x squared minus yx right square bracket equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space open square brackets left parenthesis straight z squared minus straight y squared right parenthesis space plus straight x left parenthesis straight z minus straight y right parenthesis close square brackets space equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space open square brackets left parenthesis straight z minus straight y right parenthesis thin space left parenthesis straight z plus straight y right parenthesis plus straight x space left parenthesis straight z minus straight y right parenthesis close square brackets equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z minus straight y right parenthesis thin space left square bracket straight z plus straight y plus straight x right square bracket equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight x plus straight y plus straight z right parenthesis$

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