
6th order Polynomial
Hello!
I am looking for the roots of the following equation:
$\displaystyle 2.3704s^63.5555s^41.333s^2+1$
I was told to use the substitution $\displaystyle x=s^2$ but I still can't figure out how to do it. This has to be solved in an exam as part of a bigger question and should not take longer than a minute, so I can't go for other approches.
Thanks,
Uli

Did the original equation have rational coefficients, possibly
$\displaystyle \frac{64}{27} x^3  \frac{32}{9} x^2  \frac{4}{3} x +1 = 0$ ?
If so then you can rescale by substitution of
$\displaystyle x = \frac{3}{4} y$
in the equation so it can be transformed to
$\displaystyle y^3 +2 y^2 +y 1 = 0$.
Sadly, if this is the case then there are no easy roots to find via easy analytical methods (unless you happen to know the cumbersome general solution for a cubic) so the only approach available would be to use a numerical method such as NewtonRaphson method.
So the bottom line is that this will most likely need a numerical method.
To do this I recommend that you first differentiate to find the maximum and minimum coordinates. From these values you should have a rough idea of where the single real root is. You can then put this first good guess into your NewtonRaphson iteration method to find the root.
Finally, don't forget to work backwards to get back to $\displaystyle s$.