Question

Find n if (x^n-27) is divisible by (x-3) . 1 2 3 4

Answer 1

(x^3-27)=(x^3-3^3)=(x-3)(x^2-3x+3^2)
which is divisible by {x-3}

Answer 2

${x}^{n} - 27 \\ if \: n = 3 \\ {x}^{3} - 27 \\ {x}^{3} - ( {3)}^{3} = (x - 3)( {x}^{2} + 3x + 9) \\ as \: we \: know \: the \: identity \\ {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2} ) \\ so \\ \frac{ {x}^{3} - {y}^{3} }{(x - 3)} = \frac{(x - 3)( {x}^{2} + 3x + 9)}{x - 3} \\ =( {x}^{2} + 3x + 9) \\ so \: n = 3 \:$