http://helmut.knaust.info/mediawiki/index.php?action=history&feed=atom&title=A_GeoGebra_applet 2026-04-10T13:52:57Z Revision history for this page on the wiki MediaWiki 1.19.1 http://helmut.knaust.info/mediawiki/index.php?title=A_GeoGebra_applet&diff=466&oldid=prev 2013-03-25T08:59:45Z

<p>‎<span dir="auto"><span class="autocomment">The complex roots of a quartic polynomial</span></span></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 08:59, 25 March 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>==The complex roots of a quartic polynomial==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">=</ins>==The complex roots of a quartic polynomial<ins class="diffchange diffchange-inline">=</ins>==</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; 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The function f(x) is graphed in the usual x-y plane, while the four roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;p&gt;The coordinate system is used here in two different ways. The function f(x) is graphed in the usual x-y plane, while the four roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&#160;</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">&lt;br&gt;&lt;br&gt;</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>Helmut Knaust 3/25/2013</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>Helmut Knaust 3/25/2013</div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=A_GeoGebra_applet&diff=465&oldid=prev 2013-03-25T08:57:06Z

<p>‎<span dir="auto"><span class="autocomment">The complex roots of a quartic polynomial</span></span></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 08:57, 25 March 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 2:</td> <td colspan="2" class="diff-lineno">Line 2:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;ggb_applet width=&quot;922&quot; height=&quot;463&quot;&#160; version=&quot;4.2&quot; ggbBase64=&quot;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&quot; 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The function f(x) is graphed in the usual x-y plane, while the four roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;p&gt;The coordinate system is used here in two different ways. The function f(x) is graphed in the usual x-y plane, while the four roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">&lt;p&gt;</del>Helmut Knaust 3/25/2013</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Helmut Knaust 3/25/2013</div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=A_GeoGebra_applet&diff=464&oldid=prev 2013-03-25T08:56:33Z

<p>‎<span dir="auto"><span class="autocomment">The complex roots of a quartic polynomial</span></span></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 08:56, 25 March 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 2:</td> <td colspan="2" class="diff-lineno">Line 2:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;ggb_applet width=&quot;922&quot; height=&quot;463&quot;&#160; version=&quot;4.2&quot; ggbBase64=&quot;UEsDBBQACAgIADgWeUIAAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT/LP88zLLNHQVKiu5QIAUEsHCEXM3l0aAAAAGAAAAFBLAwQUAAgICAA4FnlCAAAAAAAAAAAAAAAADAAAAGdlb2dlYnJhLnhtbN0Za4/bNvJz+isIfTgkzVomReqVs1O0QYsWtwmCJi2KO1wOlETbzMqSK0prO8iPvxlSsmVvNs0Lt8HtwyTF4bxnOCPPvtutS3KtGqPrau4xn3pEVXld6Go597p2MUm87x5/M1uqeqmyRpJF3axlO/eEH3jHc7DygwQP62LuFWFIxYIvJmqRFRORimSSLWQy4VzJlMbRosgyj5Cd0Y+q+plcK7ORuXqRr9RaXta5bC3OVdtuHk2n2+3WH6j7dbOcLpeZvzOFR4Dzysy9fvII0J0c2nILHlDKpn88vXToJ7oyraxy5RGUqtOPv7k32+qqqLdkq4t2NfdSkXpkpfRyBWJGFGSaItAGZN2ovNXXysDR0dLK3K43ngWTFe7fczNSHsTxSKGvdaGauUf9gNFYJCwM05BHPBQeqRutqraHZT3N6YBtdq3V1qHFmaUoaBqDCbTRWanm3kKWBqTS1aIBjQJDTQdL0+5LlclmWB/5YRfwCwD6jUJcIKZTAyxCesEiehFTegGWdLyMCXukrevSYqUkTMnbtySgASUXODA3BDBEkdui7hnlbgjcINwQOhjhjgsHKhyMcDCCv0fOfn0UtH9wIukgJx/LyUA+/AcbXwh6U85kJCdDId4ShtzbgRPkm1n+cRD9MnLL2A6MuoH1mwl+WH1FnykR/ySJ2Iiq84fbid7wl4FiGgQfTjH4LDkPUgbvkjIIb5HyM5U7EGXhiCjQsn/2/wZJ/lFy3qraj6AYic+J/U8gGNOTsB9i3o2sH9+nhi/G1Gw6ZMNZzxAxK4TtXbpVa4Ms8tQmJ8JICMEbxZBLQsJSGGIM4oCwkIgQliwhEY4x4Ri3gnCSEIRjnNgUFCbwIWxMRyQEXPgwdsFNuCAhJ8wmLkFAC8QmP9BJwAEiDEkIh5A6Q7I8IiKCBU+IAAYx7cWYWjicgzUQDwhnhONZFpMgIlFAYkydTGBGjRLkHZAGJKIkwqOQOyFvupwJJxLCURqIgk1t9EG5K1VuDlaxetTVpmtPdJevi2Ha1mfQRZ1f/XCmayVNO8wBCC6s47XoLrCTW/PerJSZKqG2eIFuQMi1LDHKLf5FXbVkcIHAPVs2crPSuXmh2hZOGfJaXstL2ardTwBtBgYtaXuZz1SXl7rQsvodfARRIEJyuNvRQ4e7XfDQUcnruile7A04Dtn9UzU1XojCjyIRBmnK7KdH9m4nSDnsHH8S0LPJJXq8iH3OooCn/WcIh4YtPx5vCeYoq+uDZHKnzKDKZYMR1ysfF7+YH+ry+GhT66p9Ijdt19g6DfJkgzJ9Xy1LZVVrLQ4VT36V1bsXTqfc4Xq538CKOgay5ZO6rBsC8RiEwO+yHzM3Whjk7ABFLQy1EHQwki4O+ywNLIQdMzdaKLC6Y62XlA1iMjqQ0cZmGkA+9jHrMlg/dZVuL4dFq/Oro6QI/6xbZ+Bt/bFTlOwLoZxNz/xrdqWaSpXOiyqwZFd3xrn1wTXvzTqjnst29X1V/KqWEI/PJabEFlA70CPHhcr1Gg66573qJJr1N2DVPS3UslGDhKUtjJ1i7S4d+/SNxxbVT029/qW6fgk+c4NVqI4b8CNgAhPHKXuz6SDtzOSN3qDjkgwy+JU6+mahjYT8X4xjE1RjAEluUba6RRW/XCmS1+tNqXakqevWkHpBJPmzA/o6Bxcv91W91rL0iOzaVQ2+9bMq15Cw/lHJzrRACfIAuil5Kpt8BYkXPNyW83VzZVZKtS/VriUyq68VmhesjB6AdDcl1ubgG2CYAvJBAwtgbFsD94sFLCETbeXe+AShF13lOF/c3z3AUzYrwUE8A/ud6WRJdpO9w3tBtisNasctJ5js8Y8ktpA+eVa3EA+lqWFTtqMjE6IXB+AJziwKUOTrbglyk43UjfGtdlWp1shxawO76taq0fnByZXVCdix6z0h8JPe1ODhpM5eQ3o+lkHu0NEhYf+W2Ae+NyuJ3UyfzEq5V82Jt1lsT+uip9zDmRLbILLWcC9NAkC0ljss8QBjZuqya6EVBAevjq2gi9o+h0Pph43mDmchzvaY22KcLfROHS4wcBn9BmLyNMCOaaiFi+UK2itjq9S2z4p28rMuClUdGJaHcIArYuMkJnCtIS3q89FhMOveZuRRwnDW+Us7Fed2SnzxFdmJhb2dcPLRdhKDnZzF7sRO4RcyVH5uKOqn/yeGSllvpyi9IzMxnH8JK2XnVpqIryrvDVb6eCMltDeSYHdkpOBDTZRLk6vSVUcwPoc7yzaVtvF44nZgqXabBnjq640OEO9eBQ+Db3cPwzmQIMpqcffqfvCAPIS+7VuygzEkc3JoU6djlLPpkRrMByY+j50M2MnfwU7Ws5Pfyg44U9f+BYX70EgCHotzQkTSI00fkCn2mBZ1JbGNGhkW6Jxgvk3uEbnezxfQuu2wv8OKZo50xUGWV/f5AyvPkZ+iZ6fv8k4jbyiPvCPyG1F2UnaPqs0PiTL6gVFGP9XZTzy2Xq9lVYCu8ZXFE1eA/QolmXdsmCVFGR3/Vv32yZv/QIBIhhOgJgOcwI0jOU6Eo9Njv6FC288d9IOY3q/Bd6epKLEaxCFzw6fq0DYPBpPNhPoBD8KYsgBqLMFgSGzymTA/SOMwjJNYJIJTmgqPvHHfIJw3IMY1VX0567atyNiYnrwDcE/Pupbbk/653oKvWG8BdxXQudruXmv8q9EaFNNBFAgexhHjSQAKitxN56dcRJRFEacxYyL5cYIvee9aceKrURwqKElS+0KKphiKe4fvf6GjG7dLC30362+Yv/3Z1e3f+3vGLeAiuX8pX6o//rX4N9407qn7fNcFg+i8U9x3ecFoY5k/f3NjX18aqEEXxzf+oMun/Zeq7r0m9QbN09HrlueoYoJ2DPw04mEYxSyNQs7iKLaW5IkfhmBXLtKY0TROxnYd22I6ftNiX4r235w+/i9QSwcI56kCa5UIAADpHQAAUEsBAhQAFAAICAgAOBZ5QkXM3l0aAAAAGAAAABYAAAAAAAAAAAAAAAAAAAAAAGdlb2dlYnJhX2phdmFzY3JpcHQuanNQSwECFAAUAAgICAA4FnlC56kCa5UIAADpHQAADAAAAAAAAAAAAAAAAABeAAAAZ2VvZ2VicmEueG1sUEsFBgAAAAACAAIAfgAAAC0JAAAAAA==&quot; 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showResetIcon = &quot;false&quot; enableRightClick = &quot;false&quot; errorDialogsActive = &quot;true&quot; enableLabelDrags = &quot;false&quot; showMenuBar = &quot;false&quot; showToolBar = &quot;false&quot; showToolBarHelp = &quot;false&quot; showAlgebraInput = &quot;false&quot; useBrowserForJS = &quot;true&quot; allowRescaling = &quot;true&quot; /&gt;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;p&gt;The coordinate system is used here in two different ways. The function f(x) is graphed in the usual x-y plane, while the four roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;p&gt;The coordinate system is used here in two different ways. The function f(x) is graphed in the usual x-y plane, while the four roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&lt;p&gt;Helmut Knaust 3/25/2013</ins></div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=A_GeoGebra_applet&diff=463&oldid=prev 2013-03-25T08:56:00Z

<p>‎<span dir="auto"><span class="autocomment">The complex roots of a quartic polynomial</span></span></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 08:56, 25 March 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>==The complex roots of a quartic polynomial==</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>==The complex roots of a quartic polynomial==</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;ggb_applet width=&quot;922&quot; 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The function f(x) is graphed in the usual x-y plane, while the four roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&lt;p&gt;The <ins class="diffchange diffchange-inline">coordinate system </ins>is used here in two different ways. The function f(x) is graphed in the usual x-y plane, while the four roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=A_GeoGebra_applet&diff=462&oldid=prev 2013-03-25T08:54:38Z

<p>‎<span dir="auto"><span class="autocomment">The complex roots of a quartic polynomial</span></span></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 08:54, 25 March 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>==The complex roots of a quartic polynomial==</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>==The complex roots of a quartic polynomial==</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;ggb_applet width=&quot;922&quot; 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The function f(x) is graphed in the usual x-y plane, while the roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&lt;p&gt;The plane is used here in two different ways. The function f(x) is graphed in the usual x-y plane, while the <ins class="diffchange diffchange-inline">four </ins>roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=A_GeoGebra_applet&diff=461&oldid=prev 2013-03-25T08:54:07Z

<p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 08:54, 25 March 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>==The complex roots of a quartic polynomial==</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>==The complex roots of a quartic polynomial==</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;ggb_applet width=&quot;922&quot; height=&quot;463&quot;&#160; version=&quot;4.2&quot; 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The function f(x) is graphed in the usual x-y plane, while the roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The plane is used here in two different ways. The function f(x) is graphed in the usual x-y plane, while the roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=A_GeoGebra_applet&diff=460&oldid=prev 2013-03-25T08:53:43Z

<p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 08:53, 25 March 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">==The complex roots of a quartic polynomial==</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;ggb_applet width=&quot;922&quot; height=&quot;463&quot;&#160; version=&quot;4.2&quot; ggbBase64=&quot;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&quot; showResetIcon = &quot;false&quot; enableRightClick = &quot;false&quot; errorDialogsActive = &quot;true&quot; enableLabelDrags = &quot;false&quot; showMenuBar = &quot;false&quot; showToolBar = &quot;false&quot; showToolBarHelp = &quot;false&quot; showAlgebraInput = &quot;false&quot; useBrowserForJS = &quot;true&quot; allowRescaling = &quot;true&quot; /&gt;</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;ggb_applet width=&quot;922&quot; height=&quot;463&quot;&#160; version=&quot;4.2&quot; ggbBase64=&quot;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&quot; showResetIcon = &quot;false&quot; enableRightClick = &quot;false&quot; errorDialogsActive = &quot;true&quot; enableLabelDrags = &quot;false&quot; showMenuBar = &quot;false&quot; showToolBar = &quot;false&quot; showToolBarHelp = &quot;false&quot; showAlgebraInput = &quot;false&quot; useBrowserForJS = &quot;true&quot; allowRescaling = &quot;true&quot; /&gt;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>The plane is used here in two different ways. The function f(x) is graphed in the usual x-y plane, while the roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>The plane is used here in two different ways. The function f(x) is graphed in the usual x-y plane, while the roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=A_GeoGebra_applet&diff=459&oldid=prev 2013-03-25T08:53:01Z

<p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 08:53, 25 March 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>&lt;ggb_applet width=&quot;922&quot; height=&quot;463&quot;&#160; version=&quot;4.2&quot; ggbBase64=&quot;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&quot; 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showResetIcon = &quot;false&quot; enableRightClick = &quot;false&quot; errorDialogsActive = &quot;true&quot; enableLabelDrags = &quot;false&quot; showMenuBar = &quot;false&quot; showToolBar = &quot;false&quot; showToolBarHelp = &quot;false&quot; showAlgebraInput = &quot;false&quot; useBrowserForJS = &quot;true&quot; allowRescaling = &quot;true&quot; /&gt;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">The plane is used here in two different ways. The function f(x) is graphed in the usual x-y plane, while the roots are in the complex plane. Note also that the roots - if complex - come in conjugate pairs.</ins></div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=A_GeoGebra_applet&diff=458&oldid=prev 2013-03-25T08:52:09Z

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