Question

cho a+b+c=0 chứng minh rằng (a-b)c^3+(c-a)b^3+(b-c)a^3=0

Answer 1

a+b+c=0

=>c=-a-b; b=-c-a; a=-b-c

=>c^3=-(a+b)^3; b^3=-(c+a)^3; a^3=-(b+c)^3

(a-b)*c^3+(c-a)*b^3+(b-c)*a^3

=(a-b)*c^3+(c-b+b-a)*b^3+(b-c)*a^3

=(a-b)*c^3+(b-a)*b^3+(c-b)*b^3+(b-c)*a^3

=(a-b)(c^3-b^3)+(c-b)(b^3-a^3)

=(a-b)*[-(a+b)^3+(a+c)^3]+(c-b)*[-(c+a)^3+(b+c)^3]

=(a-b)(a+c-a-b)(a^2+2ac+c^2+a^2+ac+ba+bc+a^2+2ab+b^2)+(c-b)*(b+c-c-a)(b^2+2bc+c^2+bc+c^2+ab+ac+c^2+2ac+a^2)

=(a-b)(c-b)(3a^2+b^2+c^2+3ac+3ab+bc)+(c-b)*(b-a)*(3c^2+b^2+a^2+3bc+3ac+ab)

=(a-b)(c-b)(3a^2+b^2+c^2+3ac+3ab+bc-3c^2-b^2-a^2-3bc-3ac-ab)

=(a-b)(c-b)(2a^2-2c^2+2ab-2bc)

=(a-b)(c-b)*2[(a-c)(a+c)+b(a-c)]

=(a-b)(c-b)*2*(a+b+c)

=0