F.2 Math(急!!)

a) Factorize x^3-27
x^3-27
= x^3 -3^3
= (x -3)(x^2 +3x +3^2) 公式:a^3 -b^3=(a -b)(a^2 +ab +b^2)
= (x -3)(x^2 +3x +9)

b) hence, factorize x^3-27x+54
x^3 -27x +54
= x^3 -27(1 + x-1) +54
= x^3 -27 -27(x -1)+54
= x^3 -27 -27(x -1)+27(2)
= x^3 -27 -27(x -3)
= (x -3)(x^2 +3x +9) -27(x -3) 由a)的結果
= (x -3)(x^2 +3x +9 -27)
= (x -3)(x^2 +3x -18)
= (x -3)(x -3)(x +6)
= (x -3)^2 (x +6)

a. Factorize x^3-27

(x)^3 - (3)^3
(x - 3)(x^2 + 3x + 9)

b. Factorize x^3-27x+54

x^3-9x-18x+54=0
x(x^2-9)-18(x-3)=0
x(x-3)(x+3)-18(x-3)=0
(x-3)(x^2+3x-18)=0
(x-3)(x+6)(x-3)=0