已知x=1/√3+√2.求证:x5+x4-10x3-10x2+x+1=0

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已知x=1/√3+√2.求证:x5+x4-10x3-10x2+x+1=0

已知x=1/√3+√2.求证:x5+x4-10x3-10x2+x+1=0
已知x=1/√3+√2.求证:x5+x4-10x3-10x2+x+1=0

已知x=1/√3+√2.求证:x5+x4-10x3-10x2+x+1=0
证明:∵x=1/(√3+√2)
∴x=√3-√2
x+1=1+√3-√2
x^2=5-2√6
x^4=49-20√6
∴x^5+x^4-10x^3-10x^2+x+1=(x+1)(x^4-10x^2+1)
=(1+√3-√2)[49-20√6-10(5-2√6)+1]
=(1+√3-√2)*(50-20√6-50+20√6)
=0
∴x=1/3+√2,x^5+x^4-10x^3-10x^2+x+1=0

x=√3-√2
设 y=√3+√2
则 xy=1
x5+x4-10x3-10x2+x+1=0
两边乘以y^3
x^2+x-10-10y+y^2+y^3=0
x-10y+y^3=0
两边乘以x
x^2-10+y^2=0
则 0=0x^2+x-10-10y+y^2+y^3=0 x-10y+y^3=0 两边乘以x x^2-10...

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x=√3-√2
设 y=√3+√2
则 xy=1
x5+x4-10x3-10x2+x+1=0
两边乘以y^3
x^2+x-10-10y+y^2+y^3=0
x-10y+y^3=0
两边乘以x
x^2-10+y^2=0
则 0=0

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