成法公式

x^3+y^3=(x+y)(x^2+y^2-xy)=1/3
得到：xy=2/9 ,x^5+y^5=(x+y)(x^4-x^3y+x^2y^2-xy^3+y^4),=(1-2xy)^2-(2/9)^2-xy(1-2xy)=31/81
a+b=1，a2+b2=2得到：ab=-1/2 a^7+b^7=(a+b)(a^6-a^5b+a^4b^2-a^3b^3+a^2b^4-ab^5+b^6)=a^6+b^6-ab(a^4+b^4)+a^2b^2(a^2+b^2)-a^3b^3=(a^2+b^2)((a^2+b^2)^2-3a^2b^2)+1/2((a^2+b^2)^2-2a^2b^2)+1/4*2+1/8=55/8
x-y=1，x3-y3=2得到xy=1/3 x^4+y^4=(x^2+y^2)^2-2x^2y^2=((x-y)^2+2xy)^2-2*1/9=23/9
x^5-y^5=(x-y）（x^4+x^3y+x2y^2+xy^3+y^4)=23/9+xy(x^2+y^2)+x^2y^2=29/9

1、x+y=1
x2+2xy+y2=1
x3+y3=1/3
(x+y)(x2-xy+y2）=1/3
x2-xy+y2=1/3
xy=2/9
x2+y2=5/9

x^5+y^5=(x+y)(x^4-x3y+x2y2-xy3+y^4)
=x2（x2-xy+y2）+y2（x2-xy+y2)-x2y2
=1/3(x2+y2)-(xy)2
=5/27-4/81
=11/81

2、a+b=1
a2+2ab+b2=1
a2+b2=2
ab=-1/2
a2-ab+b2=5/2

a^7+b^7=(a+b)(a^6-a^5b+a^4b2-a3b3+a2b^4-ab^5+b^6)
=a^4(a2-ab+b2）-a3b3+b^4(a2-ab+b2)
=5/2*(a^4+b^4)+1/8
=5/2*(4-1)+1/8
=61/8

3、x-y=1
x3-y3=2
x2+xy+y2=2
xy=1/3
x2+y2=5/3

x^4+y^4=(x2+y2)2-2x2y2
=(5/3)2-2*1/9
=23/9

x^5-y^5=(x-y)(x^4+x3y+x2y2+xy3+y^4)
=x^4+y^4+xy(x2+xy+y2）
=23/9+（1/3)*(5/3+1/3)
=29/9

1

x3+y3=(x+y)(x2+y2-xy)=1/3
解出：xy=2/9
x+y=1
=>(x+y)2=1
=>x2+y2+2xy=1
=>x2+y2=5/9

x^5+y^5=(x2+y2)(x3+y3)-x2y2(x+y)
將x2+y2 x3+y3 x+y (xy)2的值代入
解出 x^5+y^5=9/81
2
a+b=1，a2+b2=2，求a^7+b^7=？
解：由a+b=1兩邊平方，得
a2+b2+2ab=1
2+2ab=1
ab=-1/2
所以a2+b2=2兩邊平方，得
a^4+b^4+2a2b2=4
a^4+b^4=4-2(ab)2=4-2×(-1/2)2=7/2
另外，(a+b)(a2+b2)=a3+ab2+a2b+b3，所以
2=a3+ab2+a^2b+b3
a3+b3=2-(ab2+a2b)=2-ab(a+b)=2-ab=2-(-1/2)=5/2
所以
(a3+b3)(a^4+b^4)=a^7+a^3b^4+a^4b^3+b^7
5/2×7/2=a^7+b^7+a3b3×(a+b)
35/4=a^7+b^7+(ab)3
35/4=a^7+b^7+(-1/2)3
因此，a^7+b^7=35/4+1/8=71/8。
3
x3-y3=2 =》 （x2+xy+y2)(x-y)=2 1式
x-y=1 2式
(x-y)2=1 => x2+y2-2xy=1 3式

由 1 2 3 式 解出
xy=1/3 x2+y2=5/3

x^4+y^4=(x2+y2)2-2(xy)2
將xy x2+y2的值代入 解出
x^4+y^4=23/9

x^5+y^5=(x-y)(x^4+y^4+x^3y+xy^3+x2y2)
=(x-y)[(x^4+y^4)+xy(x2+y2)+(xy)2]
將x-y x^4+y^4 xy x2+y2的 值代入 解出 x^5+y^5=29/9