If $a+b+c=0$ prove that $ 3(a^2+b^2+c^2) \times (a^5+b^5+c^5) = 5(a^3+b^3+c^3) \times (a^4+b^4+c^4) $

Question

If $a+b+c=0$, for $a,b,c \in\mathbb R$, prove

$$ 3(a^2+b^2+c^2) \times (a^5+b^5+c^5) = 5(a^3+b^3+c^3) \times (a^4+b^4+c^4) $$

I made this question as a more difficult (higher degree) version of this question. My idea was that algebraic brute force methods are easy to distinguish from more sophisticated ones if the degree of the terms in the question is higher. The question was specifically made using the method from my answer to the linked question.

Answer 1

Let's define $$S_n=a^n+b^n+c^n$$ and consider the generating function $$F(t)=\sum_{n=0}^\infty S_nt^n=\frac1{1-at}+\frac1{1-bt}+\frac1{1-ct}.$$ Using $a+b+c=0$ gives $$(1-at)(1-bt)(1-ct)=1+pt^2+qt^3$$ for some $p$ and $q$, and $$F(t)=\frac{3+pt^2}{1+pt^2+qt^3}.$$ From this we expand as a power series $$F(t)=(3+pt^2)(1-(pt^2+qt^3)+(pt^2+qt^3)^2-\cdots) =3-2pt^2-3qt^3+2p^2t^4+5pqt^5+\cdots$$ and now you can read off this, and several other identities...

Answer 2

plug in the term $$c=-a-b$$ in the left-hand side of the equation we get $$-30 a b (a+b) \left(a^2+a b+b^2\right)^2$$ and so is the right-hand side a remark: if we compute the left-hand side minus the right-hand side and factorize, we obtain $$-(a+b+c) \left(2 a^6-2 a^5 b-2 a^5 c-a^4 b^2+4 a^4 b c-a^4 c^2+6 a^3 b^3-3 a^3 b^2 c-3 a^3 b c^2+6 a^3 c^3-a^2 b^4-3 a^2 b^3 c+6 a^2 b^2 c^2-3 a^2 b c^3-a^2 c^4-2 a b^5+4 a b^4 c-3 a b^3 c^2-3 a b^2 c^3+4 a b c^4-2 a c^5+2 b^6-2 b^5 c-b^4 c^2+6 b^3 c^3-b^2 c^4-2 b c^5+2 c^6\right)$$ and it is clear that we get zero

Answer 3

Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.

Hence, $u=0$ and $$a^2+b^2+c^2=(a+b+c)^2-2(ab+ac+bc)=-6v^2,$$ $$a^3+b^3+c^3=a^3+b^3+c^3-3abc+3abc=$$ $$=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)+3abc=3abc=3w^3,$$ $$a^4+b^4+c^4=(a^2+b^2+c^2)^2-2(a^2b^2+a^2c^2+b^2c^2)=$$ $$=36v^4-2((ab+ac+bc)^2-2abc(a+b+c))=36v^4-18v^4=18v^4$$ and $$a^5+b^5+c^5=(a^3+b^3+c^3)(a^2+b^2+c^2)-\sum_{cyc}(a^3b^2+a^3c^2)=$$ $$=(-6v^2)\cdot3w^3-(a+b+c)(a^2b^2+a^2c^2+b^2c^2)+abc(ab+ac+bc)=$$ $$=-18v^2w^3+3v^2w^3=-15v^2w^3.$$ Id est, $$3(a^2+b^2+c^2)(a^5+b^5+c^5)-5(a^3+b^3+c^3)(a^4+b^4+c^4)=$$ $$=3(-6v^2)(-15v^2w^3)-5\cdot3w^3\cdot18v^4=0.$$ Done!