Given a polynomial: $$z^7+10z^6+42z^5+96z^4+129z^3+102z^2+44z+8$$ find it's roots. I started off by using Horner's method (I believe one of the roots has to be $1$, so that's my starting point) but there's an awful lot of computing here and we aren't allowed to have a calculator during the tests. Is there a more efficient way for doing this? And also , what are the most used ways for finding polynomial roots (I believe this question was asked before, and a hyperlink would be enough). Thanks.
2026-04-06 10:53:11.1775472791
Most efficient way to find polynomial roots
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Apply Descartes rule of signs to see that there are no positive roots, and 7 negative roots. Then use the rational roots theorem to find the possibilities for rational roots. Test the possibilities by synthetic division. If you find a root, the quotient of the division will be a polynomial of lower degree and the remainder will be 0. Apply the same method to the polynomial quotient, until you have identified all rational roots.