Đáp án:
a, `x^2 - x - 2017.2018`
`= (x^2 + 2017x) - (2018x + 2017.2018)`
`= x(x + 2017) - 2018(x + 2017)`
`= (x - 2018)(x + 2017)`
b, `(a + b + c)^3 - a^3 - b^3 - c^3`
`= (a + b)^3 + c^3 + 3c(a + b)(a + b + c) - a^3 - b^3 - c^3`
`= a^3 + b^3 + 3ab(a + b) + 3c(a + b)(a + b + c) - a^3 - b^3`
`= 3ab(a + b) + 3c(a + b)(a + b + c)`
`= 3(a + b)(ab + ca + cb + c^2)`
`=3(a + b)[a(b + c) + c(b + c)]`
`=3(a + b)(b + c)(c + a)`
c, Đặt `x + y = a ; y + z = b ; z + x = c`
`=> a + b + c = 2(x + y + z)`
Ta có :
`8(x+y+z)^3 - (x+y)^3 - (y+z)^3 - (z+x)^3`
`= [2(x + y + z)]^3 - (x+y)^3 - (y+z)^3 - (z+x)^3`
`= (a + b + c)^3 - a^3 - b^3 - c^3`
`= 3(a + b)(b + c)(c + a)` ( theo câu b)
Thay ngược vào ta được :
`8(x+y+z)^3 - (x+y)^3 - (y+z)^3 - (z+x)^3 `
`= 3(x + y + y + z)(y + z + z + x)(z + x + x + y)`
`= 3(x + 2y + z)(y + 2z + x)(z + 2x + y)`
c, Đặt `2x + y = a ; 2y + z = b ; 2z + x = c`
`=> a + b + c = 3(x + y + z)`
Ta có :
`27(x+y+z)^3 - (2x+y)^3 - (2y+z)^3 - (2z+x)^3`
`= [3(x + y + z)]^3 - (2x+y)^3 - (2y+z)^3 - (2z+x)^3`
`= (a + b + c)^3 - a^3 - b^3 - c^3`
`= 3(a + b)(b + c)(c + a)` ( theo câu b)
Thay ngược vào ta được :
`27(x+y+z)^3 - (2x+y)^3 - (2y+z)^3 - (2z+x)^3`
`= 3(2x + y + 2y + z)(2y + z + 2z + x)(2z + x + 2x + y)`
`= 3(2x + 3y + z)(2y + 3z + x)(2z + 3x + y)`
Giải thích các bước giải:
