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I am trying to find the roots of a 4th-degree polynomial by hand which is actually coming from the derivative of the open-loop transfer function of a control system to find the breakaway points for the root locus. The equation is as follows:$$-3s^4-22s^3-55s^2-56s-40=0$$ I know that all the roots of this polynomial are complex numbers and there comes the problem I am facing. I've always thought to find one real root to be able to find the other roots that are complex or not and all the information I found on the internet always relies on finding or guessing some real roots but that is not possible.

My control systems lecturer said it can be done for polynomials with even degrees but I just could not find an answer and I don't know what am I supposed to do.

Jupiterian
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  • Numpy roots provides $-3.16295075+0.84376464j, -3.16295075-0.84376464j, -0.50371591+0.99523508j, -0.50371591-0.99523508j$ for the roots. Not sure that it would be easy to get those... – mathcounterexamples.net Dec 11 '21 at 16:22
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    Either you have the wrong polynomial or your lecturer is insane. – Parcly Taxel Dec 11 '21 at 16:22
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    It is technically possible to find the roots of the polynomial by hand, but it is highly unlikely that your lecturer intends for you to use such a method. Since this is from the context of control theory, perhaps you're expected to use the RH stability criterion. – Ben Grossmann Dec 11 '21 at 16:22
  • We were asked to find breakaway points for the root locus and to do so we had to take the derivative of the plant function and the polynomial that I wrote came from it. Then we just wrote the complex roots as @mathcounterexamples.net provided above without any calculus and at the end of the lecture I asked how are we expected to find these roots if asked in an exam for breakaway points he said he would ask only polynomials with even degrees but now that I am thinking he might've mean polynomials with only even degree terms which makes it way easier to solve. – Jupiterian Dec 11 '21 at 16:37

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