已知x+y=-3 x^3+y^3=-18,求x^7+y^7的值

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已知x+y=-3 x^3+y^3=-18,求x^7+y^7的值
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已知x+y=-3 x^3+y^3=-18,求x^7+y^7的值
已知x+y=-3 x^3+y^3=-18,求x^7+y^7的值

已知x+y=-3 x^3+y^3=-18,求x^7+y^7的值
x^3+y^3
=(x+y)(x^2-xy+y^2)=-18
x^2-xy+y^2=-18/(-3)=6
x+y=-3
(x+y)^2=9
x^2+y^2+2xy=9
x^2-xy+y^2=6
相减
3xy=3
xy=1
x^2+y^2=7
(x^2+y^2)^2=x^4+y^4+2x^2y^2=49
x^4+y^4=49-2(xy)^2=49-2=47
x^7+y^7
=(x^3+y^3)(x^4+y^4)-(x^3y^4+x^4y^3)
=-18*47-[(xy)^3*y+(xy)^3*x]
=-846-(x+y)
=-846+3
=-843

已知条件，是那个等式吗？

x^3+y^3
=(x+y)(x^2-xy+y^2)=-18
x^2-xy+y^2=-18/(-3)=6
x+y=-3
(x+y)^2=9
x^2+y^2+2xy=9
x^2-xy+y^2=6
相减
3xy=3
xy=1
x^2+y^2=7
(x^2+y^2)^2=x^4+y^4+2x^2y^2=49
...

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x^3+y^3
=(x+y)(x^2-xy+y^2)=-18
x^2-xy+y^2=-18/(-3)=6
x+y=-3
(x+y)^2=9
x^2+y^2+2xy=9
x^2-xy+y^2=6
相减
3xy=3
xy=1
x^2+y^2=7
(x^2+y^2)^2=x^4+y^4+2x^2y^2=49
x^4+y^4=49-2(xy)^2=49-2=47
x^7+y^7
=(x^3+y^3)(x^4+y^4)-(x^3y^4+x^4y^3)
=-18*47-[(xy)^3*y+(xy)^3*x]
=-846-(x+y)
=-846+3
=-843

收起

1.
因为X+Y=-3
所以(X+Y)^2=9=X^2+2XY+Y^2 X^2+Y^2=9-2XY
又因为X^3+Y^3=-18=(X+Y)(X^2-XY+Y^2)=3 (X^2-XY+Y^2)
=3(9-3XY)=-18
所以:XY=5
(X^3+Y^3)^2=X^6+2X^3Y^3+Y^6=X^6+Y^6+2(XY)^3=(-18)^...

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1.
因为X+Y=-3
所以(X+Y)^2=9=X^2+2XY+Y^2 X^2+Y^2=9-2XY
又因为X^3+Y^3=-18=(X+Y)(X^2-XY+Y^2)=3 (X^2-XY+Y^2)
=3(9-3XY)=-18
所以:XY=5
(X^3+Y^3)^2=X^6+2X^3Y^3+Y^6=X^6+Y^6+2(XY)^3=(-18)^2
所以 X^6+Y^6=(-18)^2-2(XY)^3
(X+Y)(X^3+Y^3)^2=3*(-18)^2
=X^7+Y^7+2X^4Y^3+XY^6+X^6Y+2X^3Y^4
= X^7+Y^7+(X^4Y^3+XY^6)+(X^4Y^3+X^3Y^4)+(X^6Y+X^3Y^4)
=X^7+Y^7+XY^3(X^3+Y^3)+X^3Y^3(X+Y)+X^3Y(X^3+Y^3)
= X^7+Y^7-18XY^3+3X^3Y^3-18X^3Y
= X^7+Y^7-18XY(X^2+Y^2)+3(XY)^3
= X^7+Y^7-18*5*(9-2*5)+3*5^3= 3*(-18)^2
所以X^7+Y^7=3*(-18)^2+18*5*(9-2*5)-3*5^3
=-507

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