When I know $a+b+c, a^2+a^2+b^2, a^3+b^3+c^3$, then how can I find the $a$ and $b$ and $ c$

Question

When I know

$$a+b+c = A$$ $$a^2+a^2+b^2 = B $$ $$a^3+b^3+c^3 = C$$

Then how can I find the $a$ and $b$ and $c$?

Notice that, more generally, Newton's Identities provides a systematic way to obtain $x_1,x_2,\dots,x_n$ as roots of a polynomial from the first $n$ power sums. — Fimpellizzeri, May 20 '16 at 05:39

score 2 · Accepted Answer · answered May 20 '16 at 05:30

2

Use $$a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)$$ to find $ab+bc+ca=P$

and $$a^3+b^3+c^3-3abc=(a+b+c)\{(a+b+c)^2-3(ab+bc+ca)\}$$ to find $abc=Q$

Then $a,b,c$ are the roots of $$t^3-At^2+Pt-Q=0$$

answered May 20 '16 at 05:30

lab bhattacharjee

1 Answers1