将多项式1+2x+3x^2+4x^3+5x^4按(x+1)幂展开.

1+2x+3x^2+4x^3+5x^4=5(x+1)⁴-16(x+1)³+21(x+1)²-12(x+1)+3
(1+2x+3x^2+4x^3+5x^4)/(x+1)=(5x³-x²+4x-2)余3,
=> 1+2x+3x^2+4x^3+5x^4=(5x³-x²+4x-2)(x+1)+3;
同理5x³-x²+4x-2=(5x²-6x+10)(x+1)-12,
5x²-6x+10=(5x-11)(x+1)+21=[5(x+1)-16](x+1)+21=5(x+1)²-16(x+1)+21,
=> 1+2x+3x^2+4x^3+5x^4=(5x³-x²+4x-2)(x+1)+3
=[(5x²-6x+10)(x+1)-12](x+1)+3
={[5(x+1)²-16(x+1)+21](x+1)-12}(x+1)+3
=5(x+1)⁴-16(x+1)³+21(x+1)²-12(x+1)+3