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I think also a3 + b3 + c3 = (a+b+c)(a2 + b2 + c2 - (ab+bc+ca))That's really the only "shortcut" you need to know. The only other one I can think of is the extension of the example you quoted using three roots i.e.
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ac)