a) 27x^3 - 27x^2 + 18x - 4 = 27x^3 - 9x^2 - 18x^2 + 6x + 12x - 4
= ( 27x^3 - 9x^2 ) - ( 18x^2 - 6x ) + ( 12x - 4 )
= 9x^2 ( 3x - 1 ) - 6x ( 3x - 1 ) + 4 ( 3x - 1 )
= ( 9x^2 - 6x + 4 ) ( 3x - 1 )
b) 2x^3 - x^2 + 5x + 3 = 2x^3 + x^2 - 2x^2 - x + 6x + 3
= ( 2x^3 + x^2 ) - ( 2x^2 + x ) + ( 6x + 3 )
= x^2 ( 2x + 1 ) - x ( 2x + 1 ) + 3 ( 2x + 1 )
= ( x^2 - x + 3 ) ( 2x+ 1 )
c) x^3 + 4x^2 - 29x + 24 = x^3 - x^2 + 5x^2 - 5x - 24x + 24
= ( x^3 - x^2 ) + ( 5x^2 - 5x ) - ( 24x - 24 )
= x^2 ( x - 1 ) + 5x ( x - 1 ) - 24 ( x - 1 )
= ( x^2 + 5x - 24 ) ( x - 1 )
= ( x^2 + 8x - 3x - 24 ) ( x -1 )
= (( x^2 + 8x ) - ( 3x + 24 )) ( x - 1 )
= ( x ( x + 8 ) - 3 ( x + 8 ) ) ( x - 1 )
= ( x - 3 ) ( x+ 8 ) ( x-1 )
