1.若x+y+z=6,x³+y³+z³=36,xyz=6,则yz+zx+xy=

题目

1.若x+y+z=6,x³+y³+z³=36,xyz=6,则yz+zx+xy=
2,若a(a-1)-(a²-b)=-20,则2分之a²+b²减去ab等于
3.已知a-b=-1,求a³+3ab-b³的值（要解答过程,三种解答方法）
4.比较A=859678245分之768901423与B=859678246分之768901424的大小（要解答过程）
5.若a,b,c都是有理数,且a+b+c=0,a³+b³+c³=0,求a的五次方+b的五次方+c的五次方的值（要解答过程）
6.已知a+a分之一=5,求a²+a²分之一的值（要解答过程）
7.已知x+y=-10,xy=16,求
x²+y²
（x+2）(y+2)

答案

1.
(x+y+z)^3=(x^2+y^2+z^2+2xy+2yz+2zx)(x+y+z)
=x^3+y^3+z^3+3x^2y+3x^2z+3xy^2+3y^2z+3xz^2+3yz^2+6xyz
=36+36+3(x^2y+x^2z+y^2x+y^2z+z^2x+z^2y)=216
x^2y+x^2z+y^2x+y^2z+z^2x+z^2y=48
x^2(6-x)+y^2(6-y)+z^2(6-z)=48
6(x^2+y^2+z^2)-(x^3+y^3+z^3)=48
x^2+y^2+z^2=14
(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)=14+2(xy+yz+zx)=36
xy+yz+zx=11
2.
a(a-1)-(a^2-b)=-20
a^2-a-a^2+b=-20
a-b=20
(a^2+b^2)/2-ab
=[(a-b)^2+2ab]/2-ab
=(a-b)^2/2+ab-ab
=20^2/2
=200
3.
a^3+3ab-b^3
=(a^3-b^3)+3ab
=(a-b)(a^2+ab+b^2)+3ab
=-a^2-ab-b^2+3ab
=-a^2+2ab-b^2
=-(a-b)^2
=-(-1)^2
=-1
4.
一般的,对于正整数m,n 且m>n
n/m-(n+1)/(m+1)
=[(n+1)m-n(m+1)]/[m(m+1)]
=(mn+m-mn-n)/[m(m+1)]
=(m-n)/[m(m+1)]>0
对于本题：
A>B
5.
a+b+c=0
c=-(a+b)
a^3+b^3+c^3=0
(a+b)(a^2-ab+b^2)+c^3=0
-c(a^2-ab+b^2)+c^3=0
c(a^2-ab+b^2-c^2)=0
c[(a+b)^2-3ab-c^2]=0
c(c^2-3ab-c^2)=0
3abc=0
a,b,c中至少有一为0,又a+b+c=0,另两数互为相反数,设为t和-t
a^5+b^5+c^5=0+t^5+(-t)^5=0
a^5+b^5+c^5=0
6.
a+1/a=5
a^2+1/a^2
=a^2+2+1/a^2-2
=(a+1/a)^2-2
=25-2
=23
7.
x^2+y^2
=(x+y)^2-2xy
=(-10)^2-2×16
=100-32
=68
(x+2)(y+2)
=xy+2(x+y)+4
=16+2(-10)+4
=0