\(x^3-y^3+z^3+3xyz\)
\(=\left(x-y\right)^3+3xy\left(x-y\right)+z^3+3xyz\)
\(=\left(x-y+z\right)\left[\left(x-y\right)^2-\left(x-y\right)z+z^2\right]+3xy\left(x-y+z\right)\)
\(=\left(x-y+z\right)\left(x^2+y^2+z^2+xy+yz-zx\right)\)
\(x^3-y^3-z^3-3xyz\)
\(=\left(x-y\right)^3+3xy\left(x-y\right)-z^3-3xyz\)
\(=\left(x-y-z\right)\left[\left(x-y\right)^2+\left(x-y\right)z+z^2\right]+3xy\left(x-y-z\right)\)
\(=\left(x-y-z\right)\left(x^2+y^2+z^2+xy-yz+zx\right)\)
\(\left(x+y+z\right)^3-x^3-y^3-z^3\)
\(=\left(y+z\right)\left[\left(x+y+z\right)^2+\left(x+y+z\right)x+x^2\right]-\left(y+z\right)\left(y^2-yz+z^2\right)\)
\(=\left(y+z\right)\left(3x^2+3xy+3yz+3zx\right)\)
\(=3\left(y+z\right)\left(x^2+xy+yz+zx\right)\)
\(=3\left(x+y\right)\left(y+z\right)\left(z+x\right)\)
1) x3 - y3 + z3 + 3xyz = (x - y)(x2 + xy + y2) + z3 + 3xyz = (x - y)3 + 3xy(x - y) + z3 + 3xyz
= (x - y + z)[(x - y)2 - z(x - y) + z2] + 3xy(x - y + z) = (x - y + z)(x2 - 2xy + y2 - xz + yz + z2 + 3xy)
= (x - y + z)(x2 + y2 + z2 + xy - xz + yz)
2) x3 - y3 - z3 - 3xyz = (x - y)(x2 + xy + y2) - z3 - 3xyz = (x - y)3 + 3xy(x - y) - z3 - 3xyz
= (x - y - z)[(x - y)2 + z(x - y) + z2] + 3xy(x - y - z) = (x - y - z)(x2 - 2xy + y2 + xz - yz + z2 + 3xy)
= (x - y - z)(x2 + y2 + z2 + xy + xz - yz)
3) (x + y + z)3 - x3 - y3 - z3 = (x + y + z - x)[(x + y + z)2 + x(x + y + z) + x2] - (y + z)(y2 - yz + z2)
= (y + z)(x2 + y2 + z2 + 2xy + 2xz + 2yz + x2 + xy + xz + x2 - y2 + yz - z2)
= (y + z)(3x2 + 3xy + 3xz + 3yz)
4) (a + b + c)3 - (A + b - c)3 - (b + c - a)3 - (c + a - b)3
= (a + b + c - a -b + c)[(a + b + c)2 + (a + b + c)(a + b - c) + (a + b - c)2] - (b + c - a + c + a - b)[(b + c - a)2 - (b + c - a)(c + a - b) + (c + a - b)2]
= 2c(a2 + b2 + c2 + 2ab + 2bc + 2ac + a2 + 2ab + b2 - c2 + a2 + b2 + c2 + 2ab - 2bc - 2ac) - 2c(a2 + b2 + c2 + 2bc - 2ac - 2ab + b2 - 2ab + a2 - c2 + a2 + b2 + c2 + 2ac - 2ab - 2bc)
= 2c(3a2 + 3b2 + c2 + 6ab - 3a2 - 3b2 - c2 + 6ab)
= 2c. 12ab = 24abc
(a+b+c)3 - (a+b-c)3 - (b+c-a)3 - (c+a-b)3
= 2c[ (a+b+c)2 + (a+b+c)(a+b-c) + (a+b-c)2 ] - 2c[ (b+c-a)2 - (b+c-a)(c+a-b) + (c+a-b)2 ]
= 2c[ 4ab + a2 - 2ab + b2 - c2 + 4ab - c2 + a2 - 2ab + b2 ]
= 4c[ (a+b)2 - c2 ]
= 4c(a+b-c)(a+b+c)
Mk bị nhầm phần d nhé
