Sum of Cubes

Algebra Level 3

If a 3 + b 3 + c 3 = 3 a b c a^{3}+b^{3}+c^{3}=3abc , and a ! = b ! = c a!=b!=c , find the value of a + b + c a+b+c .


The answer is 0.

Sinjini Das by Sinjini Das , 40, India

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4 solutions

Aaaaa Bbbbb
Apr 15, 2014

a 3 + b 3 + c 3 − 3 a b c = ( a + b + c ) ( a 2 + b 2 + c 2 − a b − b c − c a ) a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c2-ab-bc-ca) = ( a + b + c ) [ ( a − b ) 2 + ( b − c ) 2 + ( c − a ) 2 ] 2 = 0 =\frac{(a+b+c)[(a-b)^2+(b-c)^2+(c-a)^2]}{2}=0 ⇒ ( a + b + c ) = 0 \Rightarrow (a+b+c)= \boxed{0}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Or it can be a=b=c o.o

Andjela Todorovic - 7 years, 1 month ago

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No we can't say that a=b=c=0 because it is already given that a != b!= c

Sushant Samuel - 7 years, 1 month ago

Both (1,1,1) and (2,2,2) work because if a=b=c then the first statement is true and the second statement is true if a!=a, and 1!=1 and 2!=2, so they both work too.

Jacob Ronkin - 7 years, 1 month ago
Prakhar Londhe
Oct 1, 2015

yeah... 3 and 6 should also be considered as options...

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Ashtik Mahapatra
May 11, 2014

if a^3+b^3+c^3 = 0 then either a=b=c or a+b+c = 0. since there is no sense in considering the case of a=b=c over here, so a+b+c = 0.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Hey in Ncert books in class 9 I have studies tis if a+b+c=0 then a^3+b^3+c^3=3abc

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