a) x8 + x7 + 1
= x8 + x7 - x6 + x6 + 1
= x6( x2 + x +1) - [ ( x3)2 - 1]
= x6( x2 + x +1) - ( x3 - 1)( x3 + 1)
= x6( x2 + x +1) - ( x -1)( x2 + x +1)( x3 + 1)
= ( x2 + x +1)[x6 - ( x -1)( x3 + 1)]
= ( x2 + x +1)[ x6 - ( x4 + x - x3 - 1)
= ( x2 + x +1)( x6 - x4 - x + x3 - 1)
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d) x7 + x5 + 1
= x7 + x6 + x5 - x6 + 1
= x5( x2 + x +1) - [( x3)2 - 1)
= x5( x2 + x +1) - ( x3 - 1)( x3 + 1)
= x5( x2 + x +1) - ( x -1)( x2 + x + 1)( x3 + 1)
= ( x2 + x +1)[ x5 - ( x -1)( x3 + 1)]
= ( x2 + x +1)[ x5 - ( x4 + x - x3 - 1)]
= ( x2 + x +1)( x5 - x4 - x + x3 + 1)
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