Lösung 3.4:7b

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According to the factor theorem, a polynomial that has the zeros
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<center> [[Image:3_4_7b.gif]] </center>
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<math>-\text{1}+i</math>
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{{NAVCONTENT_STOP}}
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and
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<math>-\text{1}-i</math>
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must contain the factors
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<math>z-\left( -\text{1}+i \right)</math>
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and
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<math>z-\left( -\text{1}-i \right)</math>. An example of such a polynomial is
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<math>\left( z-\left( -\text{1}+i \right) \right)\left( z-\left( -\text{1}-i \right) \right)=z^{2}+2z+2</math>
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NOTE: If one wants to have all the polynomials which have only these zeros, the answer is
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<math>C\left( z+1-i \right)^{m}\left( z+1+i \right)^{n}</math>
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where
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<math>C</math>
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is a non-zero constant and
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<math>m</math>
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and
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<math>n</math>
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are positive integers.

Version vom 15:55, 28. Okt. 2008

According to the factor theorem, a polynomial that has the zeros \displaystyle -\text{1}+i and \displaystyle -\text{1}-i must contain the factors \displaystyle z-\left( -\text{1}+i \right) and \displaystyle z-\left( -\text{1}-i \right). An example of such a polynomial is


\displaystyle \left( z-\left( -\text{1}+i \right) \right)\left( z-\left( -\text{1}-i \right) \right)=z^{2}+2z+2


NOTE: If one wants to have all the polynomials which have only these zeros, the answer is


\displaystyle C\left( z+1-i \right)^{m}\left( z+1+i \right)^{n}


where \displaystyle C is a non-zero constant and \displaystyle m and \displaystyle n are positive integers.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.4:7b“