证明x³-3x²+3x-1=(x-1)³

证明x³-3x²+3x-1=(x-1)³
证明x³-3x²+3x-1=(x-1)³

证明x³-3x²+3x-1=(x-1)³
证明：
左边＝x³-3x²+3x-1
=(x³-1)-(3x²-3x)
=(x-1)(x²+x+1)-3x(x-1)
=(x-1)[(x²+x+1)-3x]
=(x-1)(x²-2x+1)
=(x-1)(x-1)²
=(x-1)³＝右边
左边＝右边得证!

反证

x^3-3x^2+3x-1 = x^3-2x^2+x -x^2+2x-1 = x(x^2-2x+1)-(x^2-2x+1) =(x-1)(x^2-2x+1) =(x-1)(x-1)^2
=(x-1)^3
(拆分配方)

(x-1)³
=（x-1)²(x-1)
=(x²-2x+1)(x-1)
=x³-2x²+x-x²+2x-1
=x³-3x²+3x-1