Question

已知x+y=-3,x³+y3³=-18，求x7+y7的值.

答案-843

Answer 1

x³+y³=(x+y)(x²-xy+y²)=(x+y)[(x+y)²-3xy]
(-3)[(-3)²-3xy]=-18
xy=1
x⁷+y⁷=(x³+y³)(x⁴+y⁴)-x⁴y³-x³y⁴
=(x³+y³)[(x²+y²)²-2x²y²]-x³y³(x+y)
=(x³+y³){[(x+y)²-2xy]²-2(xy)²}-(xy)³(x+y)
=(-18)·{[(-3)²-2·1]²-2·1²}-1³·(-3)
=-843
你给出的参考答案是正确的，结果就是-843。

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谢谢

Answer 2

x^3+y^3=(x+y)(x^2-xy+y^2)=(x+y)[(x+y)^2-3xy]=(x+y)^3-3xy(x+y)=-27+9xy=-18
所以xy=1
x^7+y^7=(x^3+y^3)(x^4+y^4)-(x^3y^4+y^3x^4)=(x^3+y^3)[(x^2+y^2)^2-2x^2y^2]-x^3y^3(y+x)=-18*[(x+y)^2-4x^2y^2]-(xy)^3*(x+y)=-18*(9-4)-1*(-3)=-87

追问

答案是-843