Expression a^3 + b^3 + c^3 – 3 abc

 
 
(a - b)^2 \; + \; (b - c)^2 \; + \; (c - a)^2

= \; 2 \, (a^2 - a \,b - a \,c + b^2 - b \, c + c^2)

= \; 1/2 \, (2 \, a - b - c)^2 \; + \; 3/2 \, (b - c)^2

 

we note that

1/2 \, (a+b+c) \, ((a - b)^2 + (b - c)^2 + (c - a)^2) \; = \; a^3 \; + \; b^3 \; + \; c^3 \; - \; 3 \, (a \,b \,c)

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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