Calculating $a_1^4+a_2^4+a_3^4$ of the roots of a polynomial

Question

We have a polynomial $f=X^3+19X^2+12X+3\in\mathbb{C}[X]$ with roots $a_1,a_2,a_3$.

What is $a_1^4+a_2^4+a_3^4$? And how do I know that these roots are all different?

Edit: How can I show that the roost are different without calculating them first?

Answer 1

Let $P_i=a_1^i+a_2^i+a_3^i$. Then use Newton's Sums:

$$P_1+19=0$$ $$P_2+19P_1+2\cdot 12=0$$ $$P_3+19P_2+12P_1+3\cdot 3=0$$

$$P_4+19P_3+12P_2+3P_1=0$$

You'll find that $P_1=-19$, $P_2=337$, $P_3=-6184$, $P_4=113509$.

Answer 2

From Vieta's Formulas, we can easily get $a_1+a_2+a_3=-19$ and $a_1a_2+a_2a_3+a_3a_1=12$.

Also, $a_1^2+a_2^2+a_3^2=(a_1+a_2+a_3)^2-2(a_1a_2+a_2a_3+a_3a_1)=361-2\times12=337$.

For the three roots, we have $x^3+19x^2+12x+3=0$ and $x^4+19x^3+12x^2+3x=0$. We add this expression for $a_1, a_2, a_3$ to get:

$$\sum a_i^3+19\sum a_i^2+12\sum a_i+9=0$$

and

$$\sum a_i^4+19\sum a_i^3+12\sum a_i^2+3\sum a_i=0$$

Substituting what we know into the first equation, we get:

$$\sum a_i^3+19(337)+12(-19)+9=0$$

Hence $\sum a_i^3=-6184$.

Substituting what we know into the second equation, we get:

$$\sum a_i^4+19(-6184)+12(337)+3(-19)=0$$

Hence $\sum a_i^4=113509$.

Also, about your question why the roots are different in this polynomial, as complex roots come in pairs (complex conjugates), since you have a degree $3$ polynomial with real coefficients, a double/multiple root can only occur when all roots are real. I would suppose that this is all the simple observations we can make to simplify the problem.

We consider a real multiple root, the derivative of the polynomial at that point would be $0$. Knowing that $3r^2+38r+12=0$, (since I think you want to avoid root finding in general), we can do long division of polynomials to get $$r^3+19r^2+12r+3=\left(\frac{x}{3}+\frac{19}{9}\right)(3r^2+38r+12)+\left(-\frac{650r}{9}-\frac{67}{3}\right)=0$$

Since, $-\frac{650r}{9}-\frac{67}{3}=0$, we can get $r=-\frac{201}{650}$. From rational root theorem, it can be seen that $r$ is not a root of the original equation.