作业帮因式分解综合题已知x-y=1求代数式x^4-xy^3-x^3y-3x^2y+3xy^2+y^4的值求证1995^2+1995^2*1996^2+1996^2是完全平方数
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因式分解综合题
已知x-y=1求代数式x^4-xy^3-x^3y-3x^2y+3xy^2+y^4的值
求证1995^2+1995^2*1996^2+1996^2是完全平方数
x^4 -xy^3 -x^3y -3x^2y +3xy^2 +y^4
=(x^4 -x^3y) +(-3x^2y +3xy^2) +(y^4-xy^3)
=x^3(x-y)-3xy(x-y)+y^3(y-x)
=x^3-3xy-y^3
=(x-y)(x^2+xy+y^2)-3xy
=x^2-2xy+y^2
=(x-y)^2
=1
1995^2+1995^2×1996^2+1996^2
=1995^2×1996^2+1995^2+(1995+1)^2
=(1995×1996)^2+1995^2+1995^2+2×1995+1
=(1995×1996)^2+2×1995^2+2×1995+1
=(1995×1996)^2+2×(1995^2+1995)+1
=(1995×1996)^2+2×(1995×1996)+1
=(1995×1996+1)^2
对的
已知x-y=1求代数式x^4-xy^3-x^3y-3x^2y+3xy^2+y^4的值
x^4-xy^3-x^3y-3x^2y+3xy^2+y^4化简
=x(x^3-y^3)-y(x^3-y^3)-3xy(x-y)
=(x^3-y^3)（x-y)-3xy(x-y)
=(x-y)(x^3-y^3-3xy)
=(x-y)(x^2+y^2+xy-3xy) ...
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已知x-y=1求代数式x^4-xy^3-x^3y-3x^2y+3xy^2+y^4的值
x^4-xy^3-x^3y-3x^2y+3xy^2+y^4化简
=x(x^3-y^3)-y(x^3-y^3)-3xy(x-y)
=(x^3-y^3)（x-y)-3xy(x-y)
=(x-y)(x^3-y^3-3xy)
=(x-y)(x^2+y^2+xy-3xy)
=(x-y)(x^2+y^2-2xy)
=(x-y)(x-y)^2
=(x-y)^3
x-y=1时代入x^4-xy^3-x^3y-3x^2y+3xy^2+y^4=1
求证1995^2+1995^2*1996^2+1996^2是完全平方数
=1995^2+1995^2*(1995+1)^2+(1995+1)^2
=1995^2+1995^2(1995^2+2*1995+1)+1995^2+2*1995+1
=1995^2(1+1995^2+2*1995+1+1)+2*1995+1
=1995^4+2*1995^3+3*1995^2+2*1995+1
=(1995^4+2*1995^2+1)+(2*1995^3+2*1995)+1995^2
=(1995^2+1)^2+2*1995*(1995^2+1)+1995^2
=(1995^2+1+1995)^2
所以：1995^2+1995^2*1996^2+1996^2是完全平方数
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