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I'm taking a class that is having me research a topic and write a report on it. The assignment is to analyze the relationship between quadratic and cubic polynomials and their discriminants.

In an attempt to get used to Maple (I've only recently begun to use it) I plotted the discriminant $b^2-4ac>0$ with the fixed value of $a=1$ using the function inequal($b^2-4\cdot c>0)$

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Visually, this makes sense to me. Equations of the form $x^2\geq 0$ has at least one real solution, but $x^2<0$ does not. Looking at the cubic discriminant (found on Wikipedia):

$$\Delta = \,b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd$$

I plotted this the same way, with the assumption that $a=1$ and $b=0$. I came up with the following:

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This one doesn't appear to be as easy to understand, however. I plotted the equation ${x}^{3}+20\,x+5$ and got the following:

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This clearly has a real root, as it crosses the x-axis. But the inequality I linked seems to imply to me that the blue area is the only area with solutions.

Given that this is my first time hearing about discriminants for anything except quadratic equations, how am I misinterpreting this data?

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  • $\begingroup$ Every cubic polynomial has at least one real root: If we assume the leading coefficient of a cubic polynomial $p(x)$ is positive, $p(x) < 0$ for sufficiently small $x$ and $p(x) > 0$ for sufficiently large $x$. So, by the I.V.T. there is some $x$ such that $p(x) = 0$. $\endgroup$ – Travis Willse Sep 29 '15 at 14:52
  • $\begingroup$ If you are using Maple then you could look at its Rootfinding:-Parametric package, see here: maplesoft.com/support/help/Maple/view.aspx?path=RootFinding/… $\endgroup$ – acer Sep 29 '15 at 20:10
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If the discriminant of a cubic is $\Delta$ and if $\Delta \lt 0$ the equation has one real root and two nonreal complex roots. There is no difficulty with the example you have used. Read further in https://en.wikipedia.org/wiki/Discriminant#Cubic.

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  • $\begingroup$ So, I suppose the graph should be read as "has complex roots" vs "has solutions". The shaded area is where there are no complex roots, but the white area does have some? $\endgroup$ – Mirrana Sep 29 '15 at 14:53
  • $\begingroup$ @agent154 One might want to specify nonreal complex roots as appropriate. $\endgroup$ – Travis Willse Sep 29 '15 at 15:01
  • $\begingroup$ Correct. The shaded area is where there are three distinct real roots and the white area is where there is one real root and two complex roots. $\endgroup$ – Paul Sundheim Sep 29 '15 at 15:01
  • $\begingroup$ As Travis says, the statement should stipulate that the complex roots are not real numbers. $\endgroup$ – Paul Sundheim Sep 29 '15 at 15:08

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