若x+y=-1,则x四次方+5x³y+x²y+8x²y²+xy²+5xy³+y四次方的值等于

若x+y=-1,则x四次方+5x³y+x²y+8x²y²+xy²+5xy³+y四次方的值等于

x⁴+5x³y+x²y+8x²y²+xy²+5xy³+y⁴
=x⁴+4x³y+6x²y²+4xy³+y⁴+x³y+2x²y²+xy³ +x²y+xy²
=(x+y)⁴+xy(x²+2xy+y²)+xy(x+y)
=(x+y)⁴+xy(x+y)²+xy(x+y)
=(-1)⁴+xy(-1)²+xy(-1)
=1+xy-xy
=1

x^4+5x^3y+x^2y+8x^2y^2+xy^2+5xy^3+y^4

=(x^4+x^3y)+(4x^3y+4x^2y^2)+(4x^2y^2+4xy^3)+(xy^3+y^4)+(x^2y+xy^2)

=-x^3-4x^2y-4xy^2-y^3-xy

=-(x^3+y^3)-4(x^2y+xy^2)-xy

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

=(x+y)^2=1

x^4+5x³y+x²y+8x²y²+xy²+5xy³+y^4 =x^4+4x^3y+6x^2y^2+4xy^3+y^4+x^3y+x^2y+2x^2y^2+xy^2+xy^3 =(x+y)^4+xy(x^2+y^2)+xy(x+y)+2x^2y^2 =1+xy[(x+y)^2-2xy]-xy+2x^2y^2 =1+xy(1-2xy)-xy+2x^2y^2 =1+xy-2x^2y^2-xy+2x^2y^2 =1