Question

【題目】分解因式：

Answer 1

【答案】(x-z)(y-z)(x-y)(x+y+z)

【解析】

去括號整理后可得x3(y-z)+y3(z-x)+z3(x-y)，由x-y=(x-z)+(z-y)，原式可變?yōu)?/span>x3(y-z)+y3(z-x)+z3[(x-z)+(z-y)]，將中括號去掉，把小括號作為整體，重新分組分解即可.

xy(x2-y2)+yz(y2-z2)+zx(z2-x2)

=x3y-xy3+y3z-yz3+z3x-zx3

=x3(y-z)+y3(z-x)+z3(x-y)

∵x-y=(x-z)+(z-y),

∴原式=x3(y-z)+y3(z-x)+z3[(x-z)+(z-y)]

=(x3-z3)(y-z)+(y3-z3)(z-x)

=(x-z)(x2+xz+z2)(y-z)+(y-z)(y2+yz+z2)(z-x)

=(x-z)(y-z)(x2+zx+z2-y2-yz-z2)

=(x-z)(y-z)(x-y)(x+y+z).