Talk:Cubic function

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The contents of Cubic equation were merged into Cubic function and they now redirect here. For the contribution history and old versions of the merged article please see its history; for its talk page, see here

Would like to rate this, but wonder if it should be merged with cubic equation? Geometry guy 01:07, 21 May 2007 (UTC)

What is the name of the shape in a Cubic Function Graph? 168.170.46.5 16:39, 11 April 2006 (UTC)

Historically, it is called a cubic parabola, which is confusing if the term parabola has always meant the curve of a second-degree polynomial in your education. The 1911 Britannica article Parabola describes many curves having the name parabola. Alternative names include cubical parabola, parabola of third degree, cubic polynomial curve, third-degree polynomial curve, and polynomial curve of third degree. Other names are variations of what I have listed but substituting order for degree, or rephrasing of third degree as of degree 3.Nicknicknickandnick 21:02, 23 May 2006 (UTC)

Contents

[edit] Root-finding Formula

When I merged from [Cubic equation|talk], I noticed each had a different set of formulae for finding the roots. I chose the one that looked simpler, but thought it best to preserve this one as well (just in case the simpler one is wrong or something!) See below: —Celtic Minstrel (talk • contribs) 20:08, 6 December 2007 (UTC)

[edit] Inconsistency in Cardano's method

It is said that

u^{3}={q\over 2}\pm \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}
u=\sqrt[3]{{q\over 2}\pm \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}}. \quad (4)

But according to the equation some lines lower

u=\sqrt[3]{-{q\over 2}\pm \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}}

so

u^{3}=-{q\over 2}\pm \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}

I think the latter is consistent with the earlier stated y^6+q y^3-{p^3\over 27}=0 when applying the quadratic ABC-formula y^3={-B\over 2A} \pm \sqrt{\left({B \over 2A}\right)^2-{C \over A}}

[edit] The formula

If we have

ax^3 + bx^2 + cx + d = a(x - x_1)(x - x_2)(x - x_3) = 0,\,

let

q = \frac{9abc - 27a^2d - 2b^3}{54a^3}

and

r = \sqrt{\left (\frac{3ac-b^2}{9a^2}\right )^3 + q^2} = \sqrt{\frac{\Delta}{108a^4}}


where Δ is the discriminant defined above. Now, let

s = \sqrt[3]{q + r}

and

t = \sqrt[3]{q - r}.

The solutions are

x_1 = s+t-\frac{b}{3a},
x_2=-\frac{1}{2}(s+t)-\frac{b}{3a}+\frac{\sqrt{3}}{2}(s-t)i,
x_3=-\frac{1}{2}(s+t)-\frac{b}{3a}-\frac{\sqrt{3}}{2}(s-t)i.

[edit] Merge

I haven't seen any discussion of the merge and so this may be a moot point that has already been demonstrated, but I would just like to point out the following inconsistency:

Linear equation

Quadratic equation

Cubic function

Quartic equation

Qunitic equation

Is there any reason why cubics should be different? asyndeton talk 20:19, 6 December 2007 (UTC)

Yes, I agree that is a little odd. This was rather a hard decision. Switch it around if you like, but it would require a little fiddling with the intro section (ie before the table of contents). The reason I chose "function" instead of "equation" is that it was easier (for me) to derive the equation from the function rather than the other way around. I agree the inconsistency is not good though. —Celtic Minstrel (talk • contribs) 20:30, 6 December 2007 (UTC)
And for the record, the only one of those that doesn't have a "function" page is Quintic:
Linear function
Quadratic function
Quartic function
Quintic function is a redirect.
—Celtic Minstrel (talk • contribs) 20:44, 6 December 2007 (UTC)
I wasn't aware of the function pages; I only came across this one when I saw the suggested merge at 'Cubic equation'. I've seen that you've proposed the merging of quartic function/equation and I think as long as we're consistent about article names, all will be fine. asyndeton talk 20:48, 6 December 2007 (UTC)

[edit] Cardano/Tartaglia Contest

I saw no source for what was previously up, and what I'm saying has a source for it, so I believe I'm justified in correcting what was said.

What was previously said was that Cardano broke his promise to Tartaglia. By modern standards, this is incorrect. The promise he made to Tartaglia was that he would not publish Tartaglia's work: Cardano published del Ferro's work. Not only that, but part of the agreement was that even if he did publish something on cubics, that he give Tartaglia a decent amount of time to at least publish his results. And I don't believe it was just a few years, he gave Tartaglia about a decade to publish his results, in which Tartaglia did nothing. Cardano even published a book about arithmetic in the meantime and sent Tartaglia a copy to show that he kept his promise. That, and the fact that he could've ignored giving Tartaglia any credit for an independent proof should be something worthwhile to say.

Later on I'll add in a few more dates and specifics from my source. Fephisto (talk) 03:20, 4 March 2008 (UTC)