Lösung 3.4:7a

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There exists a simple relation between a zero and the polynomial's factorization:
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There exists a simple relation between a zero and the polynomial's factorization: <math>z=a</math> is a zero if and only if the polynomial contains the factor <math>(z-a)</math>. (This is the meaning of the factor theorem.)
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<math>z=a\text{ }</math>
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is a zero if and only if the polynomial contains the factor
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<math>\left( z-a \right)</math>. (This is the meaning of the factor theorem.)
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If we are to have a polynomial with zeros at
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If we are to have a polynomial with zeros at <math>1</math>, <math>2</math> and <math>4</math>, the polynomial must therefore contain the factors <math>(z-1)</math>, <math>(z-2)</math> and <math>(z-4)</math>. For example,
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<math>1,\ 2</math>
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and
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<math>\text{4}</math>, the polynomial must therefore contain the factors
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<math>\left( z-1 \right),\ \left( z-2 \right)</math>
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and
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<math>\left( z-4 \right)</math>. For example,
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{{Displayed math||<math>(z-1)(z-2)(z-4) = z^3-7z^2+14z-8\,\textrm{.}</math>}}
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<math>\left( z-1 \right)\left( z-2 \right)\left( z-4 \right)=z^{3}-7z^{2}+14z-8</math>
 
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Note: It is possible to multiply the polynomial above by a non-zero constant and get another third-degree polynomial with the same roots.
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NOTE: it is possible to multiply the polynomial above by a non-zero constant and get another third-degree polynomial with the same roots.
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Version vom 14:37, 31. Okt. 2008

There exists a simple relation between a zero and the polynomial's factorization: \displaystyle z=a is a zero if and only if the polynomial contains the factor \displaystyle (z-a). (This is the meaning of the factor theorem.)

If we are to have a polynomial with zeros at \displaystyle 1, \displaystyle 2 and \displaystyle 4, the polynomial must therefore contain the factors \displaystyle (z-1), \displaystyle (z-2) and \displaystyle (z-4). For example,

\displaystyle (z-1)(z-2)(z-4) = z^3-7z^2+14z-8\,\textrm{.}


Note: It is possible to multiply the polynomial above by a non-zero constant and get another third-degree polynomial with the same roots.

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