Đáp án:
`g)24abc`
Giải thích các bước giải:
`g)(a+b+c)³-(a+b-c)³-(b+c-a)³-(c+a-b)³`
Đặt `x=a+b-c`
`y=b+c-a`
`z=c+a-b`
`⇒x+y+z=a+b-c+b+c-a+c+a-b`
`=(a-a+a)+(b+b-b)+(-c+c+c)`
`=a+b+c`
Ta có:`(a+b+c)³-(a+b-c)³-(b+c-a)³-(c+a-b)³`
`=(x+y+z)³-x³-y³-z³`
`=[(x+y)+z]³-x³-y³-z³`
`=(x+y)³+3z(x+y)²+3z²(x+y)+z³-x³-y³-z³`
`=x³+3x²y+3xy²+y³+3z(x²+2xy+y²)+3xz²+3yz²+z³-x³-y³-z³`
`=x³+3x²y+3xy²+y³+3x²z+6xyz+3y²z+3xz²+3yz²+z³-x³-y³-z³`
`=(x³-x³)+(y³-y³)+(z³-z³)+3x²y+3xy²+3x²z+6xyz+3y²z+3xz²+3yz²`
`=3x²y+3xy²+3x²z+6xyz+3y²z+3xz²+3yz²`
`=3x²y+3xy²+3x²z+3xyz+3xyz+3y²z+3xz²+3yz²`
`=3(x²y+xyz+xy²+y²z+x²z+xz²+xyz+yz²)`
`=3[(x²y+xyz)+(xy²+y²z)+(x²z+xz²)+(xyz+yz²)]`
`=3[xy(x+z)+y²(x+z)+xz(x+z)+yz(x+z)]`
`=3[(x+z)(xy+y²+xz+yz)]`
`=3{(x+z)[(xy+y²)+(xz+yz)]}`
`=3{(x+z)[y(x+y)+z(x+y)]}`
`=3(x+y)(y+z)(x+z)`
`=3(a+b-c+b+c-a)(b+c-a+c+a-b)(a+b-c+c+a-b)`
`=3.2b.2c.2a`
`=24abc`
Vậy `(a+b+c)³-(a+b-c)³-(b+c-a)³-(c+a-b)³=24abc`
