已知x=√2-1,求x^5-2x^3+6x^2-7x+2的值

题目

已知x=√2-1,求x^5-2x^3+6x^2-7x+2的值

答案

x=√2-1
x^2=3-2√2
x^5-2x^3+6x^2-7x+2
=x^5-2x^3+x+6x^2-8x+2
=x(x^4-2x^2+1)+6x^2-8x+2
=x(x^2-1)^2+6x^2-8x+2
=x(3-2√2-1)^2+6x^2-8x+2
=(√2-1)(2-2√2)^2+6x^2-12x+6+4x-4
=4(√2-1)(√2-1)^2+6(x-1)^2+4x-4
=4(√2-1)(√2-1)^2+6(√2-1-1)^2+4x-4
=4(√2-1)(3-2√2)+6(√2-2)^2+4x-4
=4(√2-1)(3-2√2)+6(4-4√2)+4x-4
=4(3√2-4-3+2√2)+6(4-4√2)+4(√2-1)-4
=4(5√2-7)+6(4-4√2)+4√2-4-4
=20√2-28+24-24√2+4√2-8
=20√2-24√2+4√2+24-28-8
=-12