"Simple" Root-Finding Algorithm

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What is an exact polynomial root finding algorithm that someone with Algebra 2 experience and who knows all the functions of a TI-nspire could implement? If none, could anyone "dumb-down" a higher level algorithm for me? I've been trying to make my own for quite a while for fun (my definition of fun is... interesting), but have been unsuccessful. The method should preferably be exact rather than approximate, but approximate will work.

Thanks in advance! If anything needs to be clarified, or you find any problems with this question, feel free to let me know! :) - JacobTDC