Phân tích đa thức thành nhân tử: x^4(y - z) + y^4(z - x) + z^4(x - y)
Phân tích đa thức thành nhân tử: x4(y − z) + y4(z − x) + z4(x − y).
x4(y − z) + y4(z − x) + z4(x − y)
= x4y − x4z + y4z − y4x + z4x − z4y
= xy(x3 − y3) − z(x4 − y4) + z4(x − y)
= xy(x − y)(x2 + xy + y2) − z(x − y)(x + y)(x2 + y2) + z4(x − y)
= (x − y)[xy(x2 + xy + y2) − z(x + y)(x2 + y2) + z4]
= (x − y)(x3y + x2y2 + xy3 − x3z − x2yz − xy2z − y3z + z4)
= (x − y)[x2y(x − z) + xy2(x − z) + y3(x − z) − z(x3 − z3)]
= (x − y)(x − z)[x2y + xy2 + y3 − z(x2 + xz + z2)]
= (x − y)(x − z)(x2y + xy2 + y3 − zx2 − xz2 − z3)
= (x − y)(x − z)[x2(y − z) + x(y2 − z2) + (y3 − z3)]
= (x − y)(x − z)[x2(y − z) + x(y − z)(y + z) + (y − z)(y2 + yz + z2)]
= (x − y)(x − z)(y − z)(x2 + y2 + z2 + xy + xz + yz).