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Question

If omegais a cube root of unity, then open vertical bar table row cell x plus 1 end cell omega cell omega to the power of 2 end exponent end cell row omega cell x plus omega to the power of 2 end exponent end cell 1 row cell omega to the power of 2 end exponent end cell 1 cell x plus omega end cell end table close vertical bar equals

  1. x to the power of 3 end exponent plus 1    
  2. x to the power of 3 end exponent plus omega    
  3. x to the power of 3 end exponent plus omega to the power of 2 end exponent    
  4. x to the power of 3 end exponent    

The correct answer is: x to the power of 3 end exponent


    capital delta equals open vertical bar table row cell x plus 1 end cell omega cell omega to the power of 2 end exponent end cell row omega cell x plus omega to the power of 2 end exponent end cell 1 row cell omega to the power of 2 end exponent end cell 1 cell x plus omega end cell end table close vertical bar
    =open vertical bar table row cell x plus 1 plus omega plus omega to the power of 2 end exponent end cell omega cell omega to the power of 2 end exponent end cell row cell x plus 1 plus omega plus omega to the power of 2 end exponent end cell cell x plus omega to the power of 2 end exponent end cell 1 row cell x plus 1 plus omega plus omega to the power of 2 end exponent end cell 1 cell x plus omega end cell end table close vertical bar,left parenthesis C subscript 1 end subscript rightwards arrow C subscript 1 end subscript plus C subscript 2 end subscript plus C subscript 3 end subscript right parenthesis
    = x open vertical bar table row 1 omega cell omega to the power of 2 end exponent end cell row 1 cell x plus omega to the power of 2 end exponent end cell 1 row 1 1 cell x plus omega end cell end table close vertical bar blank open vertical bar table row 1 a cell a to the power of 2 end exponent end cell row 1 b cell b to the power of 2 end exponent end cell row 1 c cell c to the power of 2 end exponent end cell end table close vertical bar not equal to 0
    = x left square bracket 1 left curly bracket left parenthesis x plus omega to the power of 2 end exponent right parenthesis left parenthesis x plus omega right parenthesis minus 1 right curly bracket plus omega left curly bracket 1 minus left parenthesis x plus omega right parenthesis right curly bracket plus omega to the power of 2 end exponent left curly bracket 1 minus left parenthesis x plus omega to the power of 2 end exponent right parenthesis right curly bracket right square bracket
    = x left parenthesis x to the power of 2 end exponent plus omega x plus omega to the power of 2 end exponent x plus omega to the power of 3 end exponent minus 1 plus omega minus omega x minus omega to the power of 2 end exponent plus omega to the power of 2 end exponent minus omega to the power of 2 end exponent x minus omega to the power of 4 end exponent right parenthesis
    = x to the power of 3 end exponent ,blank left parenthesis because omega to the power of 3 end exponent equals 1 right parenthesis

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