Question

Sa se arate ca daca a,b,c∈ R, atunci:
a) a^{2} + b^{2} + c^{2} + ab+bc+ca= \frac{1}{2} [(a+b)² + (b+c)² + (c+a)²]
b) a^{3} + b^{3} + c^{3} = (a+b+c)³-3 (a+b)(b+c)(c+a)
c) a^{3} + b^{3} + c^{3} - 3abc=(a+b+c) · ( a^{2} + b^{2} + c^{2} - ab - bc- ca) (Va rog mult de tot explicati-mi si mie sa inteleg ca nu stiu sa le fac)

Answer 1

]$a^{2} + b^{2} + c^{2} + ab+bc+ca= \frac{1}{2} [ (a+b)^{2} + (b+c)^{2} + (c+a)^{2} ] \\ \frac{1}{2} [(a^{2} +2ab+ b^{2} +b^{2} +2bc+c^{2} + c^{2} +2ac+ a^{2} )]\\ =\frac{1}{2} (2 a^{2} +2b^{2} +2 c^{2} +2ab+2ac+2bc) \\ \frac{1}{2}*2(a^{2} + b^{2} + c^{2} + ab+bc+ca) \\ =a^{2} + b^{2} + c^{2} + ab+bc+ca$