x4+(x+y)4+y4 分解因式

分析 这是一个二元对称式,二元对称式的基本对称式是x+y,xy任何二元对称多项式都可用x+y,xy表示,如x2+y2=(x+y)2-2xy,二元对称多项式的分解方法之一是:先将其用xy,x+y表示,再行分解.

解 ∵x4+y4

=(x+y)4-4x3y-6x2y2-4xy2

=(x+y)4-4xy(x+y)2+2x2y2.

∴原式=(x+y)4-4xy(x+y)2+2x2y2+(x+y)4

=2(x+y)4-4xy(x+y)2+2x2y2

=2[(x+y)4-2xy(x+y)2+(xy)2]

=2[(x+y)2-xy]2-2(x2+y2+xy)2,

x^4+y^4+(x+y)^4
=(x^2+y^2)^2-2x^2y^2+(x+y)^4
=[(x+y)^2-2xy]^2-2x^2y^2+(x+y)^4
=[(x+y)^2]^2-4xy(x+y)^2+4x^2y^2-2x^2y^2+(x+y)^4
=2(x+y)^4-4xy(x+y)^2+2x^2y^2
=2[(x+y)^4-2xy(x+y)^2+(xy)^2]
=2[(x+y)^2-xy]^2
=2(x^2+xy+y^2)^2