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Mathematik-Online problems:

Problem 9: Complex Roots of Polynomials

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

a)
Given the polynomials $ p(z)=z^4+a_3z^3+a_2z^2+a_1z+a_0$, with $ a_0,\,\ldots , a_4\in\ \mathbb{R}$. Proof: Is $ z\in\mathbb{C}$ a root of $ p$, then the complex conjugate $ \bar{z}$ is a root of $ p$, too.
b)
Find all roots of $ p(z)=z^4+z^3+2z^2+z+1$. Hint: $ p({\mathrm{i}})=0$.
c)
Find all roots of $ q(z)= z^5+z^4-13z^3+19z^2-68z+60$.

(Authors: Kimmerle/Roggenkamp/Höfert)

see also:

[Solutions]
  automatisch erstellt am 14. 10. 2004