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factor the expression x^6-1, using the following methods

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2 Answers

a)
x^6 - 1
= (x^3)^2 - 1
= (x^3 + 1)(x^3 - 1)
= (x+1)(x^2 - x + 1)(x-1)(x^2 + x +1)
= (x+1)(x-1)(x^2 - x +1)(x^2 + x + 1)
 
b)
x^6 - 1
= (x^2)^3 - 1
= (x^2 - 1)(x^4 + x^2 + 1)
= (x+1)(x-1)(x^4+2x^2+1 - x^2)
= (x+1)(x-1)[(x^2+1)^2 - x^2)
= (x+1)(x-1)(x^2-x+1)(x^2+x+1)
 
 
 Have to use identity
 
   X - Y = ( X - Y ) ( X +Y )!
   X- Y = ( X - Y ) ( X+ xy + y2  ) 
 
       FIRST  : (x 6  - 1 ) = ( x+ 1 ) ( x3 + 1 ) = ( x + 1 ) ( x2 - x + 1 ) ( x -1) ( x2 + x + 1 )