Question

当x＞1,比较代数式x³与x²-x+1的大小

Answer 1

x³+1=(x+1)(x²-x+1),

因此

x³+1- (x²-x+1)

=(x+1)(x²-x+1) - (x²-x+1)

=x(x²-x+1)

显然当x＞1时,x²-x+1大于1

即当x＞1时,x(x²-x+1)>1,

于是此时

x³+1- (x²-x+1) >1,

即

x³ - (x²-x+1) >0,

所以当x＞1时,

代数式 x³ > x²-x+1