Do you know your abcs?
E = ( a , b , c ) → ( 1 , 1 , 1 ) lim a 2 + b 2 + c 2 − a b − a c − b c a 3 + b 3 + c 3 − 3 a b c
Find E .
The answer is 3.00.
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1 solution
That's cheating!!! :) I intended for factoring to happen.
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Long division = factoring. a 3 + b 3 + c 3 − 3 a b c = ( a + b + c ) ( a 2 + b 2 + c 2 − a b − a c − b c ) is a common algebraic identity, which can be proven using Newton's identities .
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In the US, that's not a common identity. The way I approached a 3 + b 3 + c 3 − 3 a b c was adding and subtracting 3 a 2 b + 3 a b 2 , then factoring. But it's probably common for Brilliant members anyway.
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@Hobart Pao – We practiced long division a lot ("ad nauseam") in High School back in Switzerland, decades ago. I have not used the technique much before I started to play on Brilliant... now it often comes in handy ;)
I have watched Americans (trying to) do long division, but, in my humble opinion, they don't do it right: They draw too many weird lines ;)
Using long division, we see that the expression simplifies to a + b + c , so that the limit is 3