If a + b + c = 0 and a^3 + b^3 + c^3 = 216, what is the value of a*b*c

Question

If  a + b + c = 0  and  a^3 + b^3 + c^3 = 216 , what is the value of  a*b*c  ?

A. -72
B. 48
C. 72
D. 24
E. 216

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lacktutor · Accepted Answer

a+b +c = 0  and  a^{3}+b^{3}+c^{3}= 216. 
-->  a +b = -c

Raise both sides to the 3rd power:
 (a+b)^{3} = (-c)^{3}

a^{3} + 3a^{2}*b+ 3a*b^{2} +b^{3} = -c^{3}

a^{3}+ b^{3}+ c^{3}+ 3ab(a+b)=0

a^{3}+ b^{3}+ c^{3}+ 3ab(-c)=0

216 - 3abc =0

abc = \frac{216}{3} = 72

The answer is C.

sujoykrdatta · Answer

We know that: 
a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ac)

However, a+b+c=0

=> a³+b³+c³-3abc = 0
=> a³+b³+c³= 3abc
=> 3abc = 216

=>  abc = 72

Answer C

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Archit3110 · Answer

an alternate way would be since we are given answer options a*b*c; find the factors of given answer options also given is that a+b+c=0 which means that either the integer values are two -ve or two +ve and 1 -ve
lets start from 72; 2^3 * 3^2
which can be ; 9-8-1=0
and its cube ; 9^3-8^3-1 = 216 ; satisfies our condition
so a*b*c ; 72
IMO C

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If a + b + c = 0 and a^3 + b^3 + c^3 = 216, what is the value of abc

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