Lösung 3.4:7b

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According to the factor theorem, a polynomial that has the zeros
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According to the factor theorem, a polynomial that has the zeros <math>-1+i</math> and <math>-1-i</math> must contain the factors <math>z-(-1+i)</math> and <math>z-(-1-i)</math>. An example of such a polynomial is
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<math>-\text{1}+i</math>
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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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{{Displayed math||<math>(z-(-1+i))(z-(-1-i)) = z^2+2z+2\,\textrm{.}</math>}}
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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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NOTE: If one wants to have all the polynomials which have only these zeros, the answer is
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{{Displayed math||<math>C(z+1-i)^m(z+1+i)^n</math>}}
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where <math>C</math> is a non-zero constant and <math>m</math> and <math>n</math> are positive integers.
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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.
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Version vom 14:41, 31. Okt. 2008

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

\displaystyle (z-(-1+i))(z-(-1-i)) = z^2+2z+2\,\textrm{.}


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

\displaystyle C(z+1-i)^m(z+1+i)^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“