How to solve a $10$ degree Polynomial? (without using the normal tricks)

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The problem specifically is a $x^{10}$ polynomial over $x^9$ polynomial that we have to graph.

To graph, we need to find all the vertical intercepts so both polynomial need to be factored.

Is there any algorithm or application downloadable to a calculator to solve this problem?

The obvious way is to use synthetic division to find. But there are over $100$ combinations of possible values for each polynomial, so I was just thinking there had to be some faster way.