Question

x^3+y^3+z^3-3xyz用因式定理的思路因式分解

Answer 1

设f(x)=x^3-(3yz)x+y^3+z^3
其中y^3+z^3=(y+z)(y^2-yz+z^2)，即f(x)=x^3-(3yz)x+(y+z)(y^2-yz+z^2)
尝试得(x+(y+z))为原式的因式，因为f(-(y+z))=-(y+z)^3+(3yz)(y+z)+y^3+z^3=0
于是用大除法计算(x^3-(3yz)x+y^3+z^3)/(x+(y+z))，得到另一因式为x^2-(y+z)x+y^2+z^2-yz
最后整理得到(x+y+z)(x²+y²+z²-xy-xz-yz)

Answer 2

x^3+y^3+z^3-3xyz
=x³+3x²y+3xy²+y³ +z³ -3xyz-3xy²-3x²y
=(x+y)³+z³ -3xy(x+y+z)
=(x+y+z)[(x+y)²-(x+y)z+z²]-3xy(x+y+z)
=(x+y+z)(x²+y²+z²-xy-xz-yz)