Finding the roots of simultaneous equations like these

Question

I am unable to find the answer to this question. It would also be helpful if you post what areas of mathematics this relates to.

Answer 1

Expand $(a+b+c)^3$, giving

$$a^3+b^3+c^3+3a^2b+3ab^2+3b^2c+3bc^3+3c^2b+3cb^2+6abc\\ =3(a^2+b^2+c^2)(a+b+c)-2(a^3+b^3+c^3)+6abc.$$

$$7^3=3\cdot35\cdot7-2\cdot151+6abc$$

$$abc=15.$$

Answer 2

\begin{eqnarray*} (a+b+c)^2&=&a^2+b^2+c^2+2(ab+bc+ca) \\ a^3+b^3+c^3 -3abc&=&(a+b+c)(a^2+b^2+c^2-(ab+bc+ca)) \end{eqnarray*} Use the first equation to obtain a value for $ab+bc+ca$ and the second to obtain a value for $abc$.

Answer 3

Hint: $$(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)$$

Now use

$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-(ab+bc+ca))$$