已知:x=2-根号5,则x^4-8x^3+16x^2-x+1=

已知:x=2-根号5,则x^4-8x^3+16x^2-x+1=
已知:x=2-根号5,则x^4-8x^3+16x^2-x+1=

已知:x=2-根号5,则x^4-8x^3+16x^2-x+1=
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

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+16x^2-16x-4x^2+16x-x+1
=x²(x-2)²-4x(x-2)²-4x²+15x+1
=x²(x-2)²-4x(x-2)²-4x²+16x-...

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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+16x^2-16x-4x^2+16x-x+1
=x²(x-2)²-4x(x-2)²-4x²+15x+1
=x²(x-2)²-4x(x-2)²-4x²+16x-16-x+16+1
=x²(x-2)²-4x(x-2)²-4(x-2)²-x+17
=(x-2)²(x²-4x+4-8)-x+17
=(x-2)²[(x-2)²-8)-x+17
=5*(-3)-2+√5+17
=√5

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