题目

几道关于因式分解的数学题 有200分 急
上排位下方数字对应的指数
一
3 2 2 2 2 2
x -3px + (3p -q )x - p(p -q )
二
3 3
x (a+1)-xy(x-y)(a-b)+y (b+1)
三
4 4 4
(x+y) +x + y
四
3 3 3 3 3 3 3 3 3
xyz( x + y +z )- x y -y z - z x
五
2 2 2 2 2 2 2 2
(ab+cd)(a -b +c -d )+(ac+cd)(a + b - c -d )
六
3 3 2 4 2 2 4 2 2
(x + y ) - 4xy[x + x y + y -2xy( x - xy + y )]
不好意思
前面打错了
几道关于因式分解的数学题
上排位下方数字对应的指数
一
x^3 -3px^2 + (3p^2 -q^2 )x - p(p^2 -q^ )
二
x^3 (a+1)-xy(x-y)(a-b)+y^3 (b+1)
三
(x+y)^4 +x^4 + y^4
四
xyz( x ^3+ y^3 +z^3 )- x ^3 y^3 -y^3 z^3 - z ^3 x^3
五
(ab+cd)(a^2 -b^2 +c^2 -d^2 )+(ac+cd)(a^2 + b^2 - c^2 -d^2 )
六
(x^3 + y^3 )^2 - 4xy[x^4 + x^2 y^2 + y^4 -2xy( x^2 - xy + y^2 )]

提问时间：2020-12-14

答案

1.x^3-3px^2+(3p^2-q^2)x-p(p^2-q^2)
=x^3-3px^2+2xp^2+(p^2-q^2)x-p(p^2-q^2)
=x(x^2-3px+2p^2)+(p^2-q^2)(x-p)
=x(x-p)(x-2p)+(p^2-q^2)(x-p)
=(x-p)[x(x-2p)+(p^2-q^2)]
=(x-p)[x^2-2px+p^2-q^2]
=(x-p)[(x-p)^2-q^2]
=(x-p)(x-p+q)(x-p-q)
2.x^3(a+1)-xy(x-y)(a-b)+y^3(b+1)
=x^3(a+1)-xy(x-y)(a+1)+xy(x-y)(b+1)+y^3(b+1)
=x(a+1)(x^2-xy+y^2)+y(b+1)(x^2+y^2-xy)
=(x^2+y^2-xy)[(a+1)x+(b+1)y]
3.x^4+y^4+(x+y)^4
=x^4+y^4+x^4+y^4+4x^2y^2+4x^3y+2x^2y^2+4xy^3 =2x^4+2y^4+6x^2y^2+4x^3y+4xy^3
=2(x^4+y^4+3x^2y^2+2x^3y+2xy^3)
=2[(x^4+2x^2y^2+y^4)+(2x^3y+2xy^3+x^2y^2]
=2[(x^2+y^2)^2+2xy(x^2+y^2)+x^2y^2]
=2(x^2+y^2+xy)^2
4.xyz(x^3+y^3+z^3)-x^3y^3-y^3z^3-z^3x^3
=x^4yz+xyz(y^3+z^3)-y^3z^3-x^3(y^3+z^3)
=yz(x^4-y^2z^2)+x(yz-x^2)(y^3+z^3)
=yz(x^2+yz)(x^2-yz)-x(x^2-yz)(y^3+z^3)
=(x^2-yz)(x^2yz+y^2z^2-xy^3-xz^3)
=(x^2-yz)[y^2(z^2-xy)-xz(z^2-xy)]
=(x^2-yz)(y^2-xz)(z^2-xy)
5.(ab+cd)(a^2-b^2+c^2-d^2)+(ac+bd)(a^2+b^2-c^2-d^2)
=(ab+cd)(a^2-d^2)-(ab+cd)(b^2-c^2)+(ac+bd)(a^2-d^2)+(ac+bd)(b^2-c^2)
=(ab+cd+ac+bd)(a^2-d^2)+(ac+bd-ab-cd)(b^2-c^2)
=(a+d)(b+c)(a+d)(a-d)+(a-d)(c-b)(b+c)(b-c)
=(a-d)(b+c)[(a+d)^2-(b-c)^2]
=(a-d)(b+c)(a+d+b-c)(a+d-b+c)
6.(x^3+y^3)^2-4xy[x^4+(xy)^2+y^4-2xy(x^2-xy+y^2)]
=(x^3+y^3)^2-4xy[(x^4+2x^2y^2+y^4)-x^2y^2-2xy(x^2-xy+y^2)]
=(x^3+y^3)^2-4xy[(x^2+y^2)^2-(xy)^2-2xy(x^2-xy+y^2)]
=(x^3+y^3)^2-4xy[(x^2+y^2+xy)(x^2+y^2-xy)-2xy(x^2-xy+y^2)]
=(x^3+y^3)^2-4xy(x^2+y^2-xy)(x^2+y^2+xy-2xy)
=(x+y)^2(x^2+y^2-xy)^2-4xy(x^2+y^2-xy)^2
=(x^2+y^2+2xy-4xy)(x^2+y^2-xy)^2
=(x-y)^2(x^2+y^2-xy)^2