x-y=1,x的立方-y的立方=2,求x的平方+y的平方=?

题目

x-y=1,x的立方-y的立方=2,求x的平方+y的平方=?
还有x的四次方+y的四次方=?x的五次方-y的五次方=?

答案

x-y=1,x^3-y^3=2
x^3-y^3=(x-y)(x^2+xy+y^2)=2
x^2+xy+y^2=2
(x-y)^2-3xy=2
3xy=(x-y)^2-2=1-2=-1
xy=-1/3
x^2+y^2=(x-y)^2-2xy=1-2*(-1/3)=1+2/3=5/3
x^4+y^4=(x^2+y^2)^2-2x^2y^2=(x^2+y^2)^2-2(xy)^2=(5/3)^2-2*(-1/3)^^2=25/9-2/9=23/9
x^5-y^5=(x-y)(x^4+x^3y+x^2y^2+xy^3+y^4)
=(x-y)[(x^4+y^4)+x^2y^2(x+y)+x^2y^2]
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