What method(s) do computers or calculators use to solve polynomial equations?

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I want to make a program that will solve polynomials, but I don't want to just look at another program that already works as I think that would defeat the purpose of the exercise.

And that's why I'm asking the question here instead of on Stack Overflow.

I think it would be very difficult to implement Alpha-Beta Theory/Veita's Formula, and guessing using Factoring doesn't seem like a neat solution to me, so I'm wondering if there is in fact an easier implementation.

I would also like this to be scalable for higher degree polynomials if possible.

Thanks.