9x^4-3x^3+7x^2-3x-2
=9x^4-3x^3-2x^2+9x^2-3x-2
=x^2(9x^2-3x-2)+(9x^2-3x-2)
=(x^2-1)(9x^2-3x-2)
=(x-1)(x+1)(9x^2-3x-2)
x^2+3xy+2y^2+4x+5y+3
=(x+y)(x+2y)+3(x+y)+(x+2y+3)
=(x+y)(x+2y+3)+(x+2y+3)
=(x+y+1)(x+2y+3)
x^4-2x^3-27x^2-44x+7
设原式=(x^2+ax+1)(x^2+bx+7)或=(x^2+ax-1)(x^2+bx-7)
x^4+(a+b)x^3+(ab+8)x^2+(7a+b)x+7=x^4-2x^3-27x^2-44x+7
或x^4+(a+b)x^3+(ab-8)x^2-(7a+b)x+7=x^4-2x^3-27x^2-44x+7
可得方程组
a+b=-2
ab+8=-27
7a+b=-44
或a+b=-2
ab-8=-27
-7a-b=-44
解第一个方程组可得a=-7 b=5,第二个无解
所以原式=(x²-7x+1)(x²+5x+7)
x+y+z=3a
(x-a)+(y-a)+(z-a)=0
设x-a=m,y-a=n,z-a=p
则m+n+p=0 p=-(m+n)
[(x-a)(y-a)+(y-a)(z-a)+(z-a)(x-a)]/[(x-a)^2+(y-a)^2+(z-a)^2]
=(mn+np+pm)/(m^2+n^2+p^2)
=[mn+p(m+n)]/(m^2+n^2+p^2)
=[mn-(m+n)^2]/[m^2+n^2+(m+n)^2]
=(mn-m^2-2mn-n^2)/2(m^2-mn+n^2)
=-(m^2-mn+n^2)/2(m^2-mn+n^2)
=-1/2