檢視一元三次方程式的原始碼

[[分類:數學]]
<table class=nicetable>
<tr><th rowspan=2>n</th><th rowspan=2>1,<br/>n,<br/><span style='font-size:80%'>-(n+1)</span></th><td rowspan=2>x<sup>3</sup>-3px+q=0<br/>設 x=u+v<br/><span style='font-size:80%'>或 x=uω<sup>2</sup>+vω<br/>或 x=uω+vω<sup>2</sup></span></td><td rowspan=2><span style='font-size:80%'>設 uv=p,u<sup>3</sup>+v<sup>3</sup>=-q<br/>則 [uω,vω<sup>2</sup>] 與<br/>[uω<sup>2</sup>,vω] 兩數對的<br/>乘積=p且立方和=-q</span></td><th rowspan=2><math>Z^2+n(n+1)Z+[\frac{n(n+1)+1}{3}]^3=0</math></th><th colspan=2>
<math>Z=\frac{-n(n+1)}{2} \pm \sqrt{[\frac{n(n+1)}{2}]^2-\frac{[n(n+1)+1]^3}{3^3}}</math>
</th><th rowspan=2>u,v=<math>\frac{n}{2}</math>±<math>\frac{n+2}{2\sqrt{3}}i</math></th></tr>
<tr><th colspan=2>
<math>Z=\frac{-n(n+1)}{2} \pm \frac{(n-1)(2n+1)(n+2)}{6\sqrt{3}}i</math>
</th></tr>

<tr><th>n=ω</th><th>1,ω,ω<sup>2</sup></th><th>x<sup>3</sup>-0x-1=0</th><th>uv=0,u<sup>3</sup>+v<sup>3</sup>=1</th><th>Z<sup>2</sup>-Z=0</th><th>Z=0,1</th><th>u,v=0,1 或0,ω 或0,ω<sup>2</sup></th><th>u,v=<math>\frac{ω}{2}</math>±<math>\frac{ω+2}{2\sqrt{3}}i</math></th></tr>

<tr><th>n=-1</th><th>1,-1,0</th><th>x<sup>3</sup>-x=0</th><th>uv=1/3,u<sup>3</sup>+v<sup>3</sup>=0</th><th>Z<sup>2</sup>+<math>\frac{1}{27}</math>=0</th><th>Z=±<math>\frac{1}{3\sqrt{3}}</math>i</th><th>u,v=±<math>\frac{1}{\sqrt{3}}</math>i</th><th>uω,vω<sup>2</sup>=<br/><math>\frac{-1}{2}</math>±<math>\frac{-1+2}{2\sqrt{3}}i</math></th></tr>

<tr><th>n=0 </th><th>1,0,-1</th><th>x<sup>3</sup>-x=0</th><th>uv=1/3,u<sup>3</sup>+v<sup>3</sup>=0</th><th>Z<sup>2</sup>+<math>\frac{1}{27}</math>=0</th><th>Z=±<math>\frac{1}{3\sqrt{3}}</math>i</th><th>u,v=±<math>\frac{1}{\sqrt{3}}</math>i</th><th>u,v=<math>\frac{0}{2}</math>±<math>\frac{0+2}{2\sqrt{3}}i</math></th></tr>

<tr><th>n=1 </th><th>1,1,-2</th><th>x<sup>3</sup>-3x+2=0</th><th>uv=1,u<sup>3</sup>+v<sup>3</sup>=-2</th><th>Z<sup>2</sup>+2Z+1=0</th><th>Z=-1,-1</th><th>u,v=-1,-1 或-ω,-ω<sup>2</sup></th><th>u,v=<math>\frac{1}{2}</math>±<math>\frac{1+2}{2\sqrt{3}}i</math></th></tr>

<tr><th>n=2 </th><th>1,2,-3</th><th>x<sup>3</sup>-7x+6=0</th><th>uv=7/3,u<sup>3</sup>+v<sup>3</sup>=-6</th><th>Z<sup>2</sup>+6Z+<math>(\frac{7}{3})^3</math>=0</th><th>Z=-3±<math>\frac{10i}{3\sqrt{3}}</math></th><th>u,v=1±<math>\frac{2}{\sqrt3}i</math></th><th>u,v=<math>\frac{2}{2}</math>±<math>\frac{2+2}{2\sqrt{3}}i</math></th></tr>

<tr><th>n=3 </th><th>1,3,-4</th><th>x<sup>3</sup>-13x+12=0</th><th>uv=13/3,u<sup>3</sup>+v<sup>3</sup>=-12</th><th>Z<sup>2</sup>+12Z+<math>(\frac{13}{3})^3</math>=0</th><th>Z=-6±<math>\frac{35i}{3\sqrt{3}}</math></th><th>u,v=<math>\frac{3}{2}</math>±<math>\frac{2.5}{\sqrt3}i</math></th><th>u,v=<math>\frac{3}{2}</math>±<math>\frac{3+2}{2\sqrt{3}}i</math></th></tr>

</table>