<s>[INST] Find the equation with roots, $A$, $B$, $C$ is $ABC=6$, $A+B+C=5$ and $A^2 +B^2+C^2=21$ Find the equation with roots, $A$, $B$, $C$ is $ABC=6$, $A+B+C=5$ and $A^2 +B^2+C^2=21$
Can someone please hint me, or show me what do i do with this question please. Im quite clueless and need to be shown the method for this question
Cheers
 [/INST] You can use the Newton's Identities.
First because we want to find 3 roots, that implies that those are going to be the zeroes of a third degree polynomial. so we can write it as:
$$a_3x^3 + a_2x^2 + a_1x + a_0 = 0$$
First we'll use the following notation:
$$s_1 = A + B + C \quad s_2 = A^2 + B^2 + C^2  \quad s_3 = A^3 + B^3 + C^3$$
For the sake of simplicity we'll use $a_3 = 1$ and from Vieta's formula we can obtain that $a_0 = -6$
So from the Newton's Identities we have:
$$a_3s_1 + a_2 = 0$$
$$1\cdot 5 + a_2 = 0$$
$$a_2 = -5$$
Now we continue:
$$a_3s_2 + a_2s_1 + 2a_1 = 0$$
$$1 \cdot 21 + (-5) \cdot 5 + 2a_1 = 0$$
$$21 - 25 + 2a_1 = 0$$
$$2a_1 = 4$$
$$a_1 = 2$$
So the roots of the polynomial:
$$x^3 - 5x^2 + 2x - 6 = 0$$
will satisfy your requirements.
The solutions are $$(4.84284873681615,0.07858-1.11030 i,0.07858+1.11030 i) \text{ and its permutations}$$
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