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If a+b+c=0, prove that a^3+b^3+c^3=3abc...

If `a+b+c=0`, prove that `a^3+b^3+c^3=3abc`

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Assertion (A) : If alpha , beta, gamma are the roots of x^3 -x-1=0 then alpha^3 + beta^3 + gamma^3 =1 Reason (R ): If a +b+c=0 then a^3 + b^3 +c^3 = 3abc

If the straight lines ax + by + c= 0 , bx +cy + a = 0 and cx +ay + b =0 are concurrent , then prove that a^(3)+b^(3)+c^(3)=3abc .

If the straight lines ax+by+c=0, bx+cy+a=0 and cx+ay+b=0 are concurrent, then prove that a^(3)+b^(3)+c^(3)=3abc

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If c^(2) ne ab and the roots of (c^(2)-ab)x^(2)-2(a^(2)-bc)x+(b^(2)-ac)=0 are equal, then show that a^(3)+b^(3)+c^(3)=3abc" or "a=0

If the centroid of the triangle formed with (a , b) , (b , c) and (c , a) is O(0 , 0) then a^(3) +b^(3) + c^(3)= .....

Suppose the that quadratic equations ax^(2)+bx+c=0 and bx^(2)+cx+a=0 have a common root. Then show that a^(3)+b^(3)+c^(3)=3abc .

(a) If x + y + z=0, show that x ^(3) + y ^(3) + z ^(3)= 3 xyz. (b) Show that (a-b) ^(3) + (b-c) ^(3) + (c-a)^(3) =3 (a-b) (b-c) (c-a)