If a + b + c = 9 and
Question:

If $a+b+c=9$ and $a^{2}+b^{2}+c^{2}=35$, find the value of $\left(a^{3}+b^{3}+c^{3}-3 a b c\right)$.

 

Solution:

$a+b+c=9$

$\Rightarrow(a+b+c)^{2}=9^{2}=81$

$\Rightarrow a^{2}+b^{2}+c^{2}+2(a b+b c+c a)=81$

$\Rightarrow 35+2(a b+b c+c a)=81$

$\Rightarrow(a b+b c+c a)=23$

We know,

$\left(a^{3}+b^{3}+c^{3}-3 a b c\right)=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)$

$=(9)(35-23)$

$=108$

 

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