Verified answer

a) x^10 + x^5 + 1

= (x^10 + x^9 + x^8) - (x^9 + x^8 + x^7) + (x^7 + x^6 + x^5) - (x^6 + x^5 + x^4) + (x^5 + x^4 + x^3)

- (x^3 + x^2 + x) +( x^2 + x + 1)

= x^8(x^2 + x + 1) - x^7(x^2 + x + 1) + x^5(x^2 + x + 1) - x^4(x^2 + x + 1) + x^3(x^2 + x + 1)

-x(x^2 + x + 1) + (x^2 + x + 1)

= (x^2 + x + 1)(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)

b)x^7 + x^5 + 1

= (x^7 + x^6 + x^5) - (x^6 + x^5 + x^4) + (x^5 + x^4 + x^3) - (x^3 + x^2 + x)

+ (x^2 + x + 1)

= (x^2 + x + 1)(x^5 - x^4 + x^3 - x + 1)

c)(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

và:

x + y = 2b

y + z = 2c

x + z = 2a

(*) thành:

(x + y + z)^3 - x^3 - y^3 - z^3

= [(x + y + z) - z][(x+ y + z)^2 + x^2 + x(x+ y + z)] - (y + z)(y^2 + z^2 - yz)

= (y+z)(x^2 + y^2 + z^2 + 2xy + 2yz + 2xz + 2x^2 + xy + xz) - (y + z)(y^2 + z^2 - yz)

= (y+z)(x^2 + y^2 + z^2 + 2xy + 2yz + 2xz + 2x^2 + xy + xz - y^2 - z^2 + yz)

= (y+z)(3x^2 + 3xy + 3yz + 3xz )

= 3(y+z)(x^2 + xy + yz + xz )

= 3(y+z)[x(x+y) + z(x+y)]

= 3(x+y)(y+z)(x+z)

= 3(2c)(2b)(2a) = 24abc