乘法公式1.已知x,y,z都是正数,且x^3+y^2+z^2=3xyz,求证x=y=z2.若(a-b)^2-4(a-b)(b-c)=0求a-2b+c的值3.已知a+b+c=6,a^3+b^3+c^3=36,abc=6,求ab+bc+ac的值4.已知x+y+z=-3,x^3+y^3=-18求x^7+y^7的值5.已知x^2-x+1=0求x^10+x^20+x^30的值6.

Question

乘法公式
1.已知x,y,z都是正数,且x^3+y^2+z^2=3xyz,求证x=y=z
2.若(a-b)^2-4(a-b)(b-c)=0求a-2b+c的值
3.已知a+b+c=6,a^3+b^3+c^3=36,abc=6,求ab+bc+ac的值
4.已知x+y+z=-3,x^3+y^3=-18求x^7+y^7的值
5.已知x^2-x+1=0求x^10+x^20+x^30的值
6.若3（a^2+b^2+c^2)=(a+b+c)^2求证a=b=c

Answer 1

1)题目抄错了?
2)(a-b)^2-4(a-b)(b-c)=(a-b)(a-b-4b+4c)=(a-b)(a-5b+4c)=0
a-b=0 or a-5b+4c=0
当a-b=0,即a=b时,a-2b+c=c-b
当a-5b+4c=0,即a=5b-4c时,a-2b+c=3b-3c
3)(a+b+c)^3 = a^3+b^3+c^3+3a^2b+3ab^2+3a^2+3ac^2+3b^2+3bc^2+6abc
=a^3+b^3+c^3 +3ab(a+b)+3ac(a+c)+3bc(b+c)+6abc
=36+3ab(6-c)+3ac(6-b)+3bc(6-a)+6abc
=36-3abc+18(ab+bc+ca)=18+18(ab+bc+ca) = 6^3= 216
ab+bc+ca=11
4) 题目有错,应该是:x+y=-3,then x^2+2xy+y^2=9,x^2+y^2=9-2xy
x^3+y^3 =(x+y)(x^2-xy+y^2)=-3(9-2xy-xy)=-9(3-xy)=-18
3-xy=2,xy=1.x^2+y^2=9-2xy=9-2=7
x^4+y^4=(x^2+y^2)^2 -2x^2y^2 =7^2-2*1^2=47
(x^3+y^3)*(x^4+y^4)=x^7+y^7+ x^3y^3(x+y)=x^7+y^7+(xy)^3*(-3)=x^7+y^7-3=-18*47,
x^7+y^7=-18*47+3=-843
5)x^2-x+1=0,(x+1)(x^2-x+1)=x^3+1=0,x^3=-1
x^10+x^20+x^30=(x^3)^3 *x+(x^3)^6*x^2 +(x^3)^10
=-x +x^2 +1=0
6)3（a^2+b^2+c^2)=(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca
2a^2+2b^2+2c^2-2ab-2bc-2ca=0 = (a-b)^2+(b-c)^2+(c-a)^2
所以 a-b=0,b-c=0,c-a=0,所以a=b=c