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
 A: Hint: Completely expand the polynomial $(x - A)(x - B)(x - C)$ and collect terms. You should recognize the coefficients.
A: 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 $$(4.84284873681615,0.07858-1.11030 i,0.07858+1.11030 i) \text{ and its permutations}$$
A: I know I'm a bit late to the party, but I think I have an insightful and intuitive method.
We will use the sum and product of roots and find an expression for the coefficients of each of the terms in the cubic
As you know:
$$\alpha + \beta +\gamma = \frac {-b}{a}$$
$$\alpha \beta \gamma = \frac {-d}{a}$$
Now, applying these to the information given, $$A + B +C = \frac {-b}{a} = 5$$ and $$ABC = \frac {-d}{a} = 6$$ and for your $A^2+B^2+C^2$ when simplified gives $(\frac{b^2}{c^2})-\frac{2c}{a}=21$ but as $\frac {b}{a}$ is given $$25-\frac{2c}{a}=21$$
Simplying these down, we get 
$$d=6a, b=-5a, c=-2a$$
Because it doesn't matter what $a$ is (because we could divide through by it anyway in the final equation and nor is it possible or worthwhile to exactly figure out what it is), we can just assume it is $1$ and then we can easily compute the value of the other coefficents. The solution is $$x^3 -5x^2+2x-6$$
