# 6th order Polynomial

• May 18th 2009, 01:43 PM
Uli34
6th order Polynomial
Hello!

I am looking for the roots of the following equation:

$\displaystyle -2.3704s^6-3.5555s^4-1.333s^2+1$

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
• May 18th 2009, 02:43 PM
the_doc
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 Newton-Raphson 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 Newton-Raphson iteration method to find the root.

Finally, don't forget to work backwards to get back to $\displaystyle s$.