File talk:Graph of cubic polynomial .png

Q and P would be non zero
y = 2x3 - 3x2 - 3x + 2 = 2(x - ½)3 - 4½(x - ½) = 0|*4/(x - ½) gives (2x - 1)2 - 32 = 0 Selected cubic polynomial is in fact marginal case that, in my opinion, is not suitable to be exposed as characteristic one being easily reduced into depressed quadratic equation due to 2qQ = 9abc - 27a2d - 2b3 = 9*2*(-3)*(-3) - 27*22*2 - 2*(-3)3 = 0 and pP = b2 - 3ac = 27 ≠ 0 Therefore I would suggest you to modify the coefficients so that Q ≠ 0 for example as follows: y = x3 - 2x2 - x + 2 = (⅓)3[(3x - 2)3 - 3*7(3x - 2) + 20] = 0 is Depressed cubic for 3*1x - 2 where 2qQ = 9abc - 27a2d - 2b3 = 9*1*(-2)*(-1) - 27*12*2 - 2*(-2)3 = − 20 ≠ 0 and pP = b2 - 3ac = 7 ≠ 0 I prefer to use qQ & pP (q = ±1/0 = p) that are enabling conversion of any cubic where P ≠ 0 into Primeval form substituting $$ x = q\frac{2\sqrt{P}X}{3a}-\frac{b}{3a}  \mbox{ and  dividing depressed cubic with } 2q\sqrt{P^3}$$ $$4X^3 - p*3X =\frac{Q}{\sqrt{P^3}} = \frac{|9abc-27a^2d-2b^3|}{2\sqrt{|b^2-3ac|^3}}= T\gtreqless 1 \mbox{ where } p = \pm 1 $$ In suggested example Q = 10. q = −1, P = 7, p = +1 and T < 1 that gives: $$4X^3 - 3X =\frac{10}{\sqrt{7^3}} = \operatorname{cos}{3U}, \mbox { }x_0=\frac{-2\sqrt{7}}{3}\operatorname{cos}{\left(\frac{1}{3} \operatorname{arccos}{\frac{10}{\sqrt{7^3}} }\right)}+\frac{2}{3}= -1, $$ $$x_{1;2} = -\frac{-1-2}{2*1} \pm \sqrt{\frac{7}{3*1^2}-\frac{3}{4}\left(-1 -\frac{2}{3*1} \right)^2 }= \frac{3}{2} \pm \frac{1}{2}=2;\mbox { }1 $$ For details click http://en.wikipedia.org/wiki/Talk:Cubic_function#Chebishew_radicals_-_inconsistency_and_replacement 4X3 − 3pX = T can be bisected into Y = X3 and 4Y = ± 3X + T meaning that real roots (3 if p = +1 and 0 ≤ T ≤ 1) of any cubic converted into its Primeval form are abscissas of the points where straight line 4Y = ±3X shifted up for a quarter of T cuts basic parabola Y = X3 that is constructable. In the other words  adding movable Y = ¾X and Y = − ¾X (Line Color Red) your graph can be transformed in graphical root finding tool for all cubic where p ≠ 0. I have prepared a graph as a Word document supported by Excel Worksheet (for T and pP calculation) that can not be pasted here. If you are agreeable and interested to help me (who is unskilled in wikidrawing) in this task I will be glad to email entire article containing mentioned construction of basic cubic parabola points and Geometrical Interpretation (kind of roots Flowchart) as well. Regards PS If you agree, please, send your address as SMS on +387 62 22 71 59

77.238.216.19 (talk) 07:49, 22 September 2008 (UTC)Stap