0
$\begingroup$

I need to show that the roots of the polynomial $z^5-z^4+15$ are in the disk $|z|<2$. But how do I do this? Am I supposed to compute all the roots?

$\endgroup$
  • $\begingroup$ This looks like an exercise on Rouche's theorem. $\endgroup$ – Lord Shark the Unknown Feb 3 '18 at 6:27
  • $\begingroup$ @LordSharktheUnknown I haven't studied this theorem yet $\endgroup$ – JakeHaming Feb 3 '18 at 6:29
  • 1
    $\begingroup$ Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. $\endgroup$ – robjohn Feb 3 '18 at 6:52
3
$\begingroup$

By the triangle inequality we obtain: $$|z|^5=|z^4-15|\leq|z|^4+15,$$ which gives $|z|<2.$

Because by https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs

the equation $$x^5-x^4-15=0$$ has one positive root and since $2^5-2^4-15>0$ and $1^5-1^4-15<0$,

we see that the positive root of the last equation is later than $2$.

$\endgroup$
  • $\begingroup$ how does it give |z|<2? $\endgroup$ – JakeHaming Feb 3 '18 at 6:34
  • $\begingroup$ I added something. See now. $\endgroup$ – Michael Rozenberg Feb 3 '18 at 6:38
4
$\begingroup$

If $|z|\ge2$, then the triangle inequality says that $$ \begin{align} |z^5-z^4+15| &\ge|z|^4|z-1|-15\\[6pt] &\ge|z|^4(|z|-1)-15\\[6pt] &\ge16\cdot(2-1)-15\\[6pt] &=1 \end{align} $$ Therefore, $|z|\lt2$.

$\endgroup$
1
$\begingroup$

Note that $|-z^4+15|\leq |z|^4+15\leq 31<|z|^5$ when $|z|=2$, and so it follows from Rouche's theorem that $z^5-z^4+15$ and $z^5$ have the same number of zeros in the disk $|z|<2$, namely $5$.

$\endgroup$
0
$\begingroup$

Set $\gamma(z)=-z^4+15$ and $f(z)=z^5$. For $\vert z\vert=2\,$ we have $\vert \gamma(z)\vert \leq 2^4+15=31<\vert 2^5\vert=32=\vert f(z)\vert$. By Rouché's Theorem the number of roots of $f$ in $\vert z\vert<2\,$ $(=5)$ concides with the ones of $z^5-z^4+15\,$ in $\vert z\vert<2\,$. Therefore, $\,z^5-z^4+15\,$ has $5$ roots in $\vert z\vert<2\,$; being $5$ the total number of roots of $z^5-z^4+15\,$ in $\mathbb{C}$, then the given polynomial has all its roots in the disk $\vert z\vert \leq2$.

$\endgroup$

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.