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I want a bunch of cubics which have a turning point near the $x$-axis, both above and below the $x$-axis.
That way, the graph might not easily show whether there is a zero there, and Newton's method might give the answer.
I want a bunch so that each student gets a different one.

I know a cubic has a double zero if it shares a zero with its own derivative. If the cubic is $f(x)=x^3+ax^2+bx+c$, I think it has a double zero if $4b^3-18abc-a^2b^2+6a^3c+27c^2=0$. How can I find values of $a$, $b$, $c$ where that function of $a$, $b$ and $c$ is near zero?

Empy2
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1 Answers1

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Just fix plenty of (say integers to make your life easier) values for $a$ and $b$ and you have a quadratic in $c$. I think that's the easiest way to generate quick triples of $a,b,c$ with your desired values.

Hope that helps,

Patrick Da Silva
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