因式分解x³(x+1)(y-z)+y³(y+1)(z-x)+z³(z+1)(x-y)如题

因式分解x³(x+1)(y-z)+y³(y+1)(z-x)+z³(z+1)(x-y)如题
因式分解x³(x+1)(y-z)+y³(y+1)(z-x)+z³(z+1)(x-y)
如题

因式分解x³(x+1)(y-z)+y³(y+1)(z-x)+z³(z+1)(x-y)如题
x"'( x + 1 )( y - z ) + y"'( y + 1 )( z - x ) + z"'( z + 1 )( x - y )
= x"'( x + 1 )( y - z ) - y"'( y + 1 )( x - z ) + ( z^4 + z"' )( x - y )
这样就看到,前两个有 x"' - y"' = ( x - y )( x" + xy + y" ) ,
后一个也有 ( x - y ) ,这样就可以提取公因式 ( x - y ) ,
或者,还可以提取 ( y - z )、( x - z ) ,
这个式子分解因式,就有因式 ( x - y )( y - z )( x - z ) ,试试看吧
= ( x^4 + x"' )( y - z ) + ( y^4 + y"' )( z - x ) + ( z^4 + z"' )( x - y )
= (x^4)y + x"'y - (x^4)z - x"'z + (y^4)z + y"'z - x(y^4) - xy"' + ( z^4 + z"' )( x - y )
= (x^4)y - x(y^4) - x"'z + y"'z + x"'y - xy"' - (x^4)z + (y^4)z + ( z^4 + z"' )( x - y )
= xy( x"' - y"' ) - z( x"'- y"' ) + xy( x" - y" ) - z[ (x^4) - (y^4) ] + ( z^4 + z"' )( x - y )
= ( x"' - y"' )( xy - z ) + xy( x" - y" ) - z( x" + y" )( x" - y" ) + ( z^4 + z"' )( x - y )
= ( x - y )( x" + xy + y" )( xy - z ) + ( x - y )( x + y )( xy - x"z - y"z ) + ( z^4 + z"' )( x - y )
= ( x - y )[ ( x"'y + x"y" + xy"' - x"z - xyz - y"z ) + ( x"y - x"'z - xy"z + xy" - x"yz - y"'z ) + z^4 + z"' ]
= ( x - y )[ x"'y + x"y" + xy"' + x"y + xy" + z^4 + z"' - x"z - xyz - y"z - x"'z - x"yz - xy"z - y"'z ]
= ( x - y )[ x"'y - x"'z + x"y" - x"yz + xy"' - xy"z + x"y - x"z + xy" - xyz + z^4 - y"'z + z"' - y"z ]
= ( x - y )[ x"'( y - z ) + x"y( y - z ) + xy"( y - z ) + x"( y - z ) + xy( y - z ) - z( y"' - z"' ) - z( y" - z" ) ]
= ( x - y )( y - z )[ x"' + x"y + xy" + x" + xy - z( y" + yz + z" ) - z( y + z ) ]
= ( x - y )( y - z )[ x"' + x"y + xy" + x" + xy - y"z - yz" - z"' - yz - z" ]
= ( x - y )( y - z )[ x"' - z"' + x"y - yz" + xy" - y"z + x" - z" + xy - yz ]
= ( x - y )( y - z )[ ( x"' - z"' ) + y( x + z )( x - z ) + y"( x - z ) + ( x" - z" ) + y( x - z ) ]
= ( x - y )( y - z )( x - z )[ x" + xz + z" + xy + yz + y" + x + z + y ]
= ( x - y )( y - z )( x - z )( x + y + z + x" + y" + z" + xy + yz + xz )