For every non-constant polynomial with complex coefficients, there exists at least one complex root.
Furthermore, its degree is also the amount of its roots (with multiplicity).
Let there be a non-constant polynomial
![{\displaystyle p(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots +a_{1}z+a_{0}\quad (a_{n}\neq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c87ea4bd76a4ff58e6b29ffad3aa00ebb1741170)
Then we have
. Since the function
is continuous, there exists a
such that
.
Let us write
, for
and a polynomial
such that
.
Let
be the complex conjugate of
. Then for all
we get:
![{\displaystyle {\begin{aligned}&|p(z_{0})|^{2}\leq |p(z)|^{2}=|p(z_{0})|^{2}+|z-z_{0}|^{2m}|q(z)|^{2}+2\,{\text{Re}}\!\left[\,{\overline {p(z_{0})}}(z-z_{0})^{m}q(z)\,\right]\\[5pt]&|z-z_{0}|^{2m}|q(z)|^{2}+2\,{\text{Re}}\!\left[\,{\overline {p(z_{0})}}(z-z_{0})^{m}q(z)\,\right]\geq 0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/538d2edadbee5831f33846251ef55b791a70ad8c)
Let
for
:
![{\displaystyle {\begin{aligned}&r^{2m}{\bigl |}q(z_{0}+re^{\theta i}){\bigr |}^{2}+2\,{\text{Re}}\!\left[\,{\overline {p(z_{0})}}\,r^{m}e^{m\theta i}q(z_{0}+re^{\theta i})\,\right]\geq 0\\[5pt]&r^{m}{\bigl |}q(z_{0}+re^{\theta i}){\bigr |}^{2}+2\,{\text{Re}}\!\left[\,{\overline {p(z_{0})}}q(z_{0}+re^{\theta i})e^{m\theta i}\,\right]\geq 0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/138aa089a4aecb64afc787d4015490baa0d1899c)
Taking the limit as
yields:
![{\displaystyle {\text{Re}}\!\left[\,{\overline {p(z_{0})}}q(z_{0})e^{m\theta i}\,\right]\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a820aaff87dc7e77af523c7aadbcef4933b941d)
Let
and
.
By plugging
into the inequality and by de Moivre's formula, we get:
![{\displaystyle {\begin{aligned}&{\text{Re}}(\pm c)=\pm \,{\text{Re}}(c)\geq 0\\&{\text{Re}}(\pm ci)=\mp \,{\text{Im}}(c)\geq 0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5659f54d7d76c588317010dd6c8d24d30c4024e3)
hence
and so
.
Therefore, from
we get
.
- McDougal Algebra 2
- Holt Algebra 2
- Lial, Hornspy, Schenider Precalculus
- Alvin Ling (starter)