[tex]a) (x-y)^{3}+(y-z)^{3}+(z-x)^{3}
\\b) (a+b+c)^{2}+(a-b+c)^{2}-4b^{2}
\\c) a(b^{2}-c^{2})-b(c^{2}-a^{2})+c(a^{2}-b^{2})
\\d) x^{3}+y^{3}+z^{3}-3xyz[/tex]
a) $(x-y)^3+(y-z)^3+(z-x)^3$
$=(x-y+y-z)[(x-y)^2-(x-y)(y-z)+(y-z)^2]-(x-z)^3$
$=(x-z)(x^2-2xy+y^2-xy+xz+y^2-yz+y^2-2yz+z^2)-(x-z)(x^2-2xz+z^2)$
$=(x-z)(x^2+3y^2+z^2-3xy-3yz+xz-x^2+2xz-z^2)$
$=(x-z)(3y^2-3xy+3xz-3yz)$
$=3(x-z)[-y(x-y)+z(x-y)]$
$=3(x-y)(y-z)(z-x)$
b) $(a+b+c)^2+(a-b+c)^2-4b^2$
$=(a+c)^2+2b(a+c)+b^2+(a+c)^2-2b(a+c)+b^2-4b^2$
$=2(a+c)^2-2b^2=2[(a+c)^2-b^2]$
$=2(a+b+c)(a-b+c)$
c) $a(b^2 - c^2) - b(c^2 - a^2) + c(a^2 - b^2)$
$= ab^2 - ac^2 - bc^2 + a^2b + c(a^2-b^2) $
$ = (a^2b+ab^2)-(ac^2+bc^2)+c(a-b)(a+b)$
$=ab(a+b)-c^2(a+b)+(ac-bc)(a+b)$
$=(a+b)(ab-c^2+ac-bc)$
$=(a+b)[a(b+c)-c(b+c)]$
$=(a+b)(b+c)(a-c)$
d) $x^3+y^3+z^3-3xyz$
$=(x+y)^3+z^3-3xy(x+y)-3xyz$
$=(x+y+z)[(x+y)^2-z(x+y)+z^2)-3xy(x+y+z)$
$=(x+y+z)(x^2+2xy+y^2-xz-yz+z^2-3xy)$
$=(x+t+z)(x^2+y^2+z^2-xy-xz-yz)$
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
