已知x=2-根号5,求x^4-8x^3+16x^2-x+1的值

已知x=2-根号5,求x^4-8x^3+16x^2-x+1的值
已知x=2-根号5,求x^4-8x^3+16x^2-x+1的值

已知x=2-根号5,求x^4-8x^3+16x^2-x+1的值
x=2-√5,x-2=-√5.
x^4-8x^3+16x^2-x+1
=(x^4-4x^3+4x^2)-4x^3+12x^2-x+1
=x^2(x-2)^2-4x^3+12x^2-x+1
=5x^2-4x^3+12x^2-x+1
=-4x^3+17x^2-x+1
=-4x^3+16x^2-16x+x^2+15x+1
=-4x(x-2)^2+x^2+15x+1
=-20x+x^2+15x+1
=x^2-5x+1
=9-4√5-5(2-√5)+1
=√5