Prove that:
$open vertical bar table row straight x cell space space space straight x squared end cell cell space space 1 plus px cubed end cell row straight y cell space space straight y squared end cell cell space space 1 plus py cubed end cell row straight z cell space space straight z squared end cell cell space space 1 plus pz cubed end cell end table close vertical bar space equals space left parenthesis 1 plus pxyz right parenthesis thin space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis$
where p is any scalar.

Solution

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

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