搜狗做题网已知x+y=-1求：x^4+5x^3y+x^2y+8x^2y^2+xy^2+5xy^3+y^4的值.
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已知x+y=-1求：x^4+5x^3y+x^2y+8x^2y^2+xy^2+5xy^3+y^4的值.
x^4+5x^3y+x^2y+8x^2y^2+xy^2+5xy^3+y^4 ①
=(x+y)(x^3+4x^2y+xy+4xy^2+y^3) ②
从第①步到第②步是怎么做的?
x^4+5x^3y+x^2y+8x^2y^2+xy^2+5xy^3+y^4 ①
=(x+y)(x^3+4x^2y+xy+4xy^2+y^3) ②
从第①步到第②步是怎么做的?
x^4+5x^3y+x^2y+8x^2y^2+xy^2+5xy^3+y^4
=x^4+5x^3y+x^2y+(10x^2y^2-x^2y^2)+xy^2+5xy^3+y^4
=(5x^3+5xy^3+10x^2y^2)+(x^4+y^4-2x^2y^2)+xy^2+x^2y
=5xy(x+y)^2+(x^2-y^2)^2+xy^2+x^2y
=5xy(x+y)^2+(x+y)^2(x-y)^2+xy(x+y)
=(x+y)(x^3+4x^2y+xy+4xy^2+y^3)
=-x^3-y^3-xy-4x^2y-4xy^2
=-(x+y)(x^2-xy+y^2)-xy-4xy(x+y)
=x^2-xy+y^2-xy+4xy
=(x+y)^2
=1
=x^4+5x^3y+x^2y+(10x^2y^2-x^2y^2)+xy^2+5xy^3+y^4
=(5x^3+5xy^3+10x^2y^2)+(x^4+y^4-2x^2y^2)+xy^2+x^2y
=5xy(x+y)^2+(x^2-y^2)^2+xy^2+x^2y
=5xy(x+y)^2+(x+y)^2(x-y)^2+xy(x+y)
=(x+y)(x^3+4x^2y+xy+4xy^2+y^3)
=-x^3-y^3-xy-4x^2y-4xy^2
=-(x+y)(x^2-xy+y^2)-xy-4xy(x+y)
=x^2-xy+y^2-xy+4xy
=(x+y)^2
=1
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