已知x=根号2-1,计算x^10+2x^9-2x^8-2x^7+x^6+3x^2+5x+1的值已知x=(根号2)-1,计算x^10+2x^9-2x^8-2x^7+x^6+3x^2+5x+1的值,请写出具体过程,

问题描述:

已知x=根号2-1,计算x^10+2x^9-2x^8-2x^7+x^6+3x^2+5x+1的值
已知x=(根号2)-1,计算x^10+2x^9-2x^8-2x^7+x^6+3x^2+5x+1的值,请写出具体过程,

9

已知x=根号2-1,所以X=1
x^10+2x^9-2x^8-2x^7+x^6+3x^2+5x+1
= 10x+18x-16x-14x+6x+6x+5x+1
=x(10+18-16-14+6+6+5)+1
=x^15+1
=16
或者直接代入式中计算

x^10+2x^9-2x^8-2x^7+x^6+3x^2+5x+1
=x^10+x^9+x^9-x^8-x^8-x^7-x^7+x^6+3x^2+5x+1
=(x^10+x^9)+(x^9-x^8)-(x^8+x^7)-(x^7-x^6)+3x^2+5x+1
=x^9(x+1)+x^8(x-1)-x^7(x+1)-x^6(x-1)+3x^2+5x+1
=x^9(根号2)+x^8(根号2-2)-x^7(根号2)-x^6(根号2-2)+3x^2+5x+1
=x^8[(根号2-1)(根号2)+根号2-2]-x^6[(根号2-1)(根号2)+根号2-2]+3x^2+5x+1
=3x^2+5x+1
=5-根号2