x的平方+x-1=0 求证(x+1)的立方-(x-1)的立方=8-6x

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x的平方+x-1=0 求证(x+1)的立方-(x-1)的立方=8-6x
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x的平方+x-1=0 求证(x+1)的立方-(x-1)的立方=8-6x
x的平方+x-1=0 求证(x+1)的立方-(x-1)的立方=8-6x

x的平方+x-1=0 求证(x+1)的立方-(x-1)的立方=8-6x
(x+1)³-(x-1)³
=[(x+1)-(x-1)][(x+1)²+(x+1)(x-1)+(x-1)²]
=2(3x²+1)
由已知 x²+x-1=0得 x²=1-x
代入上式得
原式=2(3x²+1)
=2(3-3x+1)
=8-6x

∵(X+1)³-(X-1)³=(X+1-X+1)[(X+1)²+(X+1)(X-1)+(X-1)²]
=2(X²+2X+1+X²-1+X²-2X+1)
=2(3X²+1)
...

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∵(X+1)³-(X-1)³=(X+1-X+1)[(X+1)²+(X+1)(X-1)+(X-1)²]
=2(X²+2X+1+X²-1+X²-2X+1)
=2(3X²+1)
=6X²+2
∵X²+X-1=0
∴X²=1-X
∴6X²+2=6(1-X)+2
=6-6X+2
=8-6X

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