Bài `39:`
Đặt `{(a + b - c = z ),(b + c -a = x ),(c + a -b = y):}`
Suy ra :
`x + y + z = (a + b - c) + (b + c - a) + (c + a - b) = a + b + c`
Khi đó ta có :
`B = (x+ y+z)^3 - x^3 - y^3 - z^3`
`= [ (x+y) + z]^3 - x^3 - y^3 - z^3`
`= [ (x+y)^3 + 3 . (x+y)^2 . z + 3 . (x+y) . z^2 + z^3 ] - x^3 - y^3 - z^3`
`= ( x^3 + 3 x^2y + 3 xy^2 + y^3 + 3 x^2 z + 6 xyz + 3 y^2z + 3z^2x + 3z^2y + z^3)- x^3 - y^3 - z^3`
`= 3 x^2y + 3 xy^2 + 3 x^2 z + 6 xyz + 3 y^2z + 3z^2x + 3z^2y`
` = 3 ( x^2y + xy^2 + x^2 z + 2xyz + y^2 z + xz^2 + z^2y)`
` = 3 [ (x^2y + x^2z + xy^2 + xyz) + (xyz + xz^2 + y^2z + yz^2)]`
`= 3 [ x (xy + xz + y^2 + yz) + z (xy + xz + y^2 + yz)]`
`= 3 [ (xy + xz + y^2+ yz)(x+z)]`
`= 3 [ x (y+z) + y(y+z)](x+z)`
`= 3 (x+y)(y+z)(x+z)`
Ta lại có :
`{(x + y = (b + c-a) + (c + a-b) = 2c ),( y + z = (c + a-b) + (a + b -c) = 2a ),(x + z= (b + c -a)+(a+b-c) = 2b):}`
Do đó :
`B = 3 . 2c . 2a . 2b`
`B = 24abc`
Vậy `B = 24abc`
Bài `38:`
`A = a (b + c -a)^2 + b (c + a - b)^2 + c (a + b - c)^2 + (b+c-a)(c+a-b)(a+b-c)`
Đặt `{( b + c - a = x ),(c + a - b =y ),(a + b - c = z):}`
Suy ra : `x + y + z = (b + c - a) + (c + a - b) + (a+b-c) = a + b + c`
Ta lại có :
`{(x + y = (b+c-a) + (c+a-b) = 2c ),(y + z = (c + a - b) + (a + b - c) = 2a ),(x + z = (b + c -a) + (a+b-c) = 2b):}`
`=> {(c = (x+y)/2 ),( a= (y+z)/2 ),(b = (x+z)/2 ):}`
Khi đó ta có :
`A = (y+z)/2 . x^2 + (x+z)/2 . y^2 + (x+y)/2 . z^2 + xyz`
`= (x^2 (y+z) + y^2 (x+z) + z^2 (x+y) + 2xyz)/2`
`= (x^2y + x^2z + y^2x + y^2z + xz^2 + yz^2 + 2xyz)/2`
`= ( (x^2y + x^2z + xy^2 + xyz) + (xyz + xz^2 + zy^2 + yz^2) )/2`
`= ( x (xy + xz + y^2 + yz) + z (xy + xz + y^2 + yz))/2`
`= ((xy + xz + y^2 + yz)(x+z))/2`
`= ((x (y+z) + y (y+z))(x+z))/2`
` = ((x+y)(y+z)(x+z))/2`
Mà `{(x + y = 2c ),(y + z = 2a ),(x + z = 2b):}`
`=> A = (2c . 2a . 2b)/2`
`=> A = 4abc`
Vậy `A = 4abc`
Bài `40:`
`C = ab (a+b) + bc (b+c) + ac (c+a) - a^3 - b^3 - c^3 - 2abc`
`= a^2b + ab^2 + b^2c + bc^2 + ac^2 + a^2c - a^3 - b^3 - c^3 - 2abc`
`= (ab^2- b^3 + b^2c) - (ac^2 - bc^2 - c^3) - (a^3 - a^2b + a^2c) + (2a^2c - 2abc + 2ac^2)`
`= b^2 (a-b+c) - c^2 (a-b+c) - a^2 (a-b+c) + 2ac (a-b+c)`
` = (b^2 - c^2 - a^2 + 2ac)(a-b+c)`
`= [ (ab + b^2 - bc) + (ac + bc - c^2) - (a^2 + ab - ac)] (a-b+c)`
` = [ b (a+b-c) + c (a+b-c) - a (a+b-c)] (a-b+c)`
`= (b+c-a)(a+b-c)(a-b+c)`
