Question

show that
$open vertical bar table row 1 cell space space space space straight x end cell cell space space space space straight x squared end cell row cell straight x squared end cell cell space space 1 end cell cell space space straight x end cell row straight x cell space space straight x squared end cell cell space space 1 end cell end table close vertical bar space equals space left parenthesis 1 minus x cubed right parenthesis squared$

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Solution

straight L. straight H. straight S space equals space open vertical bar table row 1 cell space space space space straight x end cell cell space space space space straight x squared end cell row cell straight x squared end cell cell space space 1 end cell cell space space straight x end cell row straight x cell space space straight x squared end cell cell space space 1 end cell end table close vertical bar space equals space open vertical bar table row cell 1 plus straight x plus straight x squared end cell cell space space straight x end cell cell space space straight x squared end cell row cell 1 plus straight x plus straight x squared end cell cell space space 1 end cell cell space space straight x end cell row cell 1 plus straight x plus straight x squared end cell cell space space space straight x squared end cell cell space 1 end cell end table close vertical bar space comma space by space straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3 space space space space space space space space space space space equals space left parenthesis 1 plus straight x plus straight x squared right parenthesis space space open vertical bar table row 1 cell space space straight x end cell cell space space straight x squared end cell row 1 1 cell space straight x end cell row 1 cell space space straight x squared end cell cell space 1 end cell end table close vertical bar space space space space space space space space space space space equals space space space left parenthesis 1 plus straight x plus straight x squared right parenthesis space open vertical bar table row 1 straight x cell straight x squared end cell row 0 cell space space 1 minus straight x end cell cell space space straight x minus straight x squared end cell row cell 0 space end cell cell space space straight x squared minus straight x end cell cell space 1 minus straight x squared end cell end table close vertical bar space comma space by space straight R subscript 2 minus straight R subscript 1 space comma space straight R subscript 3 minus straight R subscript 1 space space space space space space space space space equals space space space left parenthesis 1 plus straight x plus straight x squared right parenthesis space space open vertical bar table row 1 straight x row cell negative straight x end cell cell 1 plus straight x end cell end table close vertical bar space equals space left parenthesis 1 plus straight x plus straight x squared right parenthesis space left parenthesis 1 minus straight x right parenthesis squared space space open square brackets table row cell 1 plus straight x plus straight x squared end cell end table close square brackets space space space space space space space space space equals space open square brackets table row cell left parenthesis 1 minus straight x right parenthesis left parenthesis 1 plus straight x plus straight x squared end cell end table close square brackets squared space equals space left parenthesis 1 minus straight x cubed right parenthesis squared space space space space space space space space space equals straight R. straight H. straight S space

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