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x^8+x^4+1(x+y+z)^5-x^5-y^5-z^5x^5-x^3*y^2-12*x*y^4(a^2-3*a+2)*x^2+(2*a^2-4*a+1)*x*y+(a^2-a)*y^2
题目详情
x^8+x^4+1 (x+y+z)^5-x^5-y^5-z^5 x^5-x^3*y^2-12*x*y^4 (a^2-3*a+2)*x^2+(2*a^2-4*a+1)*x*y+(a^2-a)*y^2
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答案和解析
请你在以后的提问中,将各题分开,以免造成阅读错误而无法解答.
第一题:
x^8+x^4+1
=(x^8+2x^4+1)-x^4=(x^4+1)^2-x^4=[(x^4+1)+x^2][(x^4+1)-x^2]
=(x^4+x^2+1)(x^4-x^2+1)
第二题:
(x+y+z)^5-x^5-y^5-z^5
=[(x+y)+z]^5-z^5-(x^5+y^5)
=(x+y)^5+5(x+y)^4z+10(x+y)^3z^2+10(x+y)^2z^3+5(x+y)z^4+z^5-z^5
-(x+y)(x^4-x^3y+x^2y^2-xy^3+y^4)
∴[(x+y+z)^5-x^5-y^5-z^5]/(x+y)
=(x+y)^4+5(x+y)^3z+10(x+y)^2z^2+10(x+y)z^3+5z^4
-(x^4-x^3y+x^2y^2-xy^3+y^4)
=(x^4+4x^3y+6x^2y^2+4xy^3+y^4)-(x^4-x^3y+x^2y^2-xy^3+y^4)
+5z[(x+y)^3+z^3]+10(x+y)z^2(x+y+z)
=5(x^3y+x^2y^2+xy^3)+5z(x+y+z)[(x+y)^2-(x+y)z+z^2]
+10(x+y)(y+z)z^2+10(x+y)xz^2
∴[(x+y+z)^5-x^5-y^5-z^5]/[5(x+y)]
=x^3y+x^2y^2+xy^3+z(x+y+z)(x^2+2xy+y^2-xz-yz+z^2)
+2(x+y)(y+z)z^2+2(x+y)xz^2
=x^3y+x^2y^2+xy^3+2(x+y)xz^2+xz(x^2+2xy+y^2-xz-yz+z^2)
+z(y+z)(x^2+2xy+y^2-xz-yz+z^2)+2(x+y)(y+z)z^2
=x^3y+x^2y^2+xy^3+2x^2z^2+2xyz^2+x^3z+2x^2yz+xy^2z-x^2z^2-xyz^2+xz^3
+z(y+z)(x^2+2xy+y^2-xz-yz+z^2)+2(x+y)(y+z)z^2
=x^3y+x^2y^2+xy^3+xyz^2+2x^2z^2+x^3z+2x^2yz+xy^2z-x^2z^2+xz^3
+z(y+z)(x^2+2xy+y^2-xz-yz+z^2)+2(x+y)(y+z)z^2
=(x^3y+x^3z)+(x^2y^2-x^2z^2)+(xy^3+xz^3)+(xyz^2+xy^2z)+(2x^2z^2+2x^2yz)
+z(y+z)(x^2+2xy+y^2-xz-yz+z^2)+2(x+y)(y+z)z^2
=x^3(y+z)+x^2(y+z)(y-z)+x(y+z)(y^2-yz+z^2)+xyz(y+z)+2x^2z(y+z)
+z(y+z)(x^2+2xy+y^2-xz-yz+z^2)+2(x+y)(y+z)z^2
∴[(x+y+z)^5-x^5-y^5-z^5]/[5(x+y)(y+z)]
=x^3+x^2(y-z)+x(y^2-yz+z^2)+xyz+2x^2z+z(x^2+2xy+y^2-xz-yz+z^2)
+2(x+y)z^2
=x^3+x^2y-x^2z+xy^2-xyz+xz^2+xyz+2x^2z+x^2z+2xyz+y^2z-xz^2-yz^2+z^3
+2xz^2+2yz^2
=x^3+x^2y+xy^2+2x^2z+2xyz+y^2z+yz^2+z^3+2xz^2
=(x^3+z^3)+(x^2y+2xyz+yz^2)+(xy^2+y^2z)+(2x^2z+2xz^2)
=(x+z)(x^2-xz+z^2)+y(x^2+2xz+z^2)+y^2(x+z)+2xz(x+z)
∴[(x+y+z)^5-x^5-y^5-z^5]/[5(x+y)(y+z)(x+z)]
=x^2-xz+z^2+y(x+z)+y^2+2xz=x^2+y^2+z^2+xy+yz+xz
∴(x+y+z)^5-x^5-y^5-z^5=5(x+y)(y+z)(x+z)(x^2+y^2+z^2+xy+yz+xz)
第三题:
x^5-x^3y^2-12xy^4
=x(x^4-x^2y^2-12y^4)=x(x^2-4y^2)(x^2+3y^2)=x(x-2y)(x+2y)(x^2+3y^2)
第四题:
(a^2-3a+2)x^2+(2a^2-4a+1)xy+(a^2-a)y^2
=(a-2)(a-1)x^2+[(a-1)(a-1)+a(a-2)]xy+a(a-1)y^2
=[(a-2)x+(a-1)y][(a-1)x+ay]
注:若原题不是我所表述的样子,则请你补充说明.
第一题:
x^8+x^4+1
=(x^8+2x^4+1)-x^4=(x^4+1)^2-x^4=[(x^4+1)+x^2][(x^4+1)-x^2]
=(x^4+x^2+1)(x^4-x^2+1)
第二题:
(x+y+z)^5-x^5-y^5-z^5
=[(x+y)+z]^5-z^5-(x^5+y^5)
=(x+y)^5+5(x+y)^4z+10(x+y)^3z^2+10(x+y)^2z^3+5(x+y)z^4+z^5-z^5
-(x+y)(x^4-x^3y+x^2y^2-xy^3+y^4)
∴[(x+y+z)^5-x^5-y^5-z^5]/(x+y)
=(x+y)^4+5(x+y)^3z+10(x+y)^2z^2+10(x+y)z^3+5z^4
-(x^4-x^3y+x^2y^2-xy^3+y^4)
=(x^4+4x^3y+6x^2y^2+4xy^3+y^4)-(x^4-x^3y+x^2y^2-xy^3+y^4)
+5z[(x+y)^3+z^3]+10(x+y)z^2(x+y+z)
=5(x^3y+x^2y^2+xy^3)+5z(x+y+z)[(x+y)^2-(x+y)z+z^2]
+10(x+y)(y+z)z^2+10(x+y)xz^2
∴[(x+y+z)^5-x^5-y^5-z^5]/[5(x+y)]
=x^3y+x^2y^2+xy^3+z(x+y+z)(x^2+2xy+y^2-xz-yz+z^2)
+2(x+y)(y+z)z^2+2(x+y)xz^2
=x^3y+x^2y^2+xy^3+2(x+y)xz^2+xz(x^2+2xy+y^2-xz-yz+z^2)
+z(y+z)(x^2+2xy+y^2-xz-yz+z^2)+2(x+y)(y+z)z^2
=x^3y+x^2y^2+xy^3+2x^2z^2+2xyz^2+x^3z+2x^2yz+xy^2z-x^2z^2-xyz^2+xz^3
+z(y+z)(x^2+2xy+y^2-xz-yz+z^2)+2(x+y)(y+z)z^2
=x^3y+x^2y^2+xy^3+xyz^2+2x^2z^2+x^3z+2x^2yz+xy^2z-x^2z^2+xz^3
+z(y+z)(x^2+2xy+y^2-xz-yz+z^2)+2(x+y)(y+z)z^2
=(x^3y+x^3z)+(x^2y^2-x^2z^2)+(xy^3+xz^3)+(xyz^2+xy^2z)+(2x^2z^2+2x^2yz)
+z(y+z)(x^2+2xy+y^2-xz-yz+z^2)+2(x+y)(y+z)z^2
=x^3(y+z)+x^2(y+z)(y-z)+x(y+z)(y^2-yz+z^2)+xyz(y+z)+2x^2z(y+z)
+z(y+z)(x^2+2xy+y^2-xz-yz+z^2)+2(x+y)(y+z)z^2
∴[(x+y+z)^5-x^5-y^5-z^5]/[5(x+y)(y+z)]
=x^3+x^2(y-z)+x(y^2-yz+z^2)+xyz+2x^2z+z(x^2+2xy+y^2-xz-yz+z^2)
+2(x+y)z^2
=x^3+x^2y-x^2z+xy^2-xyz+xz^2+xyz+2x^2z+x^2z+2xyz+y^2z-xz^2-yz^2+z^3
+2xz^2+2yz^2
=x^3+x^2y+xy^2+2x^2z+2xyz+y^2z+yz^2+z^3+2xz^2
=(x^3+z^3)+(x^2y+2xyz+yz^2)+(xy^2+y^2z)+(2x^2z+2xz^2)
=(x+z)(x^2-xz+z^2)+y(x^2+2xz+z^2)+y^2(x+z)+2xz(x+z)
∴[(x+y+z)^5-x^5-y^5-z^5]/[5(x+y)(y+z)(x+z)]
=x^2-xz+z^2+y(x+z)+y^2+2xz=x^2+y^2+z^2+xy+yz+xz
∴(x+y+z)^5-x^5-y^5-z^5=5(x+y)(y+z)(x+z)(x^2+y^2+z^2+xy+yz+xz)
第三题:
x^5-x^3y^2-12xy^4
=x(x^4-x^2y^2-12y^4)=x(x^2-4y^2)(x^2+3y^2)=x(x-2y)(x+2y)(x^2+3y^2)
第四题:
(a^2-3a+2)x^2+(2a^2-4a+1)xy+(a^2-a)y^2
=(a-2)(a-1)x^2+[(a-1)(a-1)+a(a-2)]xy+a(a-1)y^2
=[(a-2)x+(a-1)y][(a-1)x+ay]
注:若原题不是我所表述的样子,则请你补充说明.
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