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Lösung 3.4:7a

Aus Online Mathematik Brückenkurs 2

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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,
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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{{Displayed math||<math>(z-1)(z-2)(z-4) = z^3-7z^2+14z-8\,\textrm{.}</math>}}
+
{{Abgesetzte Formel||<math>(z-1)(z-2)(z-4) = z^3-7z^2+14z-8\,\textrm{.}</math>}}
Note: It is possible to multiply the polynomial above by a non-zero constant and get another third-degree polynomial with the same roots.
Note: It is possible to multiply the polynomial above by a non-zero constant and get another third-degree polynomial with the same roots.

Version vom 13:16, 10. Mär. 2009

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

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

(z−1)(z−2)(z−4)=z3−7z2+14z−8.


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.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.4:7a“