5
$\begingroup$

I need to calculate the sums

$$x_1^3 + x_2^3 + x_3^3$$

and

$$x_1^4 + x_2^4 + x_3^4$$

where $x_1, x_2, x_3$ are the roots of

$$x^3+2x^2+3x+4=0$$

using Viete's formulas.

I know that $x_1^2+x_2^2+x_3^2 = -2$, as I already calculated that, but I can't seem to get the cube of the roots. I've tried

$$(x_1^2+x_2^2+x_3^2)(x_1+x_2+x_3)$$

but that did work.

$\endgroup$
11
$\begingroup$

If $x_1,x_2,x_3$ are the roots of $x^3+2x^2+3x+4=0$ then $$x^3+2x^2+3x+4 = (x-x_1)(x-x_2)(x-x_3) $$ $$= x^3 - (x_1 + x_2 + x_3)x^2 + (x_1 x_2 + x_1 x_3 + x_2 x_3)x - x_1 x_2 x_3 = x^3 - e_1 x^2 + e_2 x - e_3.$$ So $e_1 = -2$, $e_2 = 3$ and $e_3 = -4$.

Now the trick is to express the power sums $x_1^3 + x_2^3 = x_3^3$ and $x_1^4 + x_2^4 = x_3^4$ in terms of the elementary symmetric polynomials $\{x_1 + x_2 + x_3,x_1 x_2 + x_1 x_3 + x_2 x_3,x_1 x_2 x_3\}$.

See my answer to the question here for details on how to do that Three-variable system of simultaneous equations

In the case of the fourth power sums you should get $x_1^4 + x_2^4 + x_3^4 = e_1^4 - 4 e_1^2 e_2 + 4 e_1 e_3 + 2 e_2^2 = 18$.

$\endgroup$
4
  • $\begingroup$ I've read the details but there is no explanation how you reached the end result, I'm more interested in that $\endgroup$ – andrei Apr 2 '11 at 13:12
  • 1
    $\begingroup$ solved it.the trick is that after you solve x1^2+x2^2+x3^2 then u procede to write (x1^2+x2^2+x3^2)*(x1+x2+x3) and get x1,2,3^3 from that equation.(you'll end up having some terms like x1^2x2 and x1^2x3 do not replace them with the first equation the sum of the roots, use the second equation and third) $\endgroup$ – andrei Apr 2 '11 at 14:37
  • $\begingroup$ Fluent and concise answer, +1!! $\endgroup$ – awllower Jan 2 '12 at 14:40
  • $\begingroup$ +1 very nice and interesting that your last line was also part of one of my questions, despite I had $4$ variables and an additional $-4e_4$. Why is this missing in your case? Ah, I see $x_4=0$. Still nice. $\endgroup$ – draks ... Apr 8 '12 at 18:34
2
$\begingroup$

I think what you need is Newton's identities, in particular the section about their application to the roots of a polynomial.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.