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How would I factor to solve for x?

$x^5 + 20x^2 + 5=0 $?

Do I use synthetic division? Is there a faster/easier way?

Do I have to keep plugging in numbers to see if they equal to zero?

Thanks! I'm not asking for full solutions if you don't want to share (but that would be nice) just opinions on what I should do.

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1 Answer

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It’s irreducible over the rational numbers, by the Eisenstein Criterion. So in particular, it doesn’t have a rational root. You can get an approximate root by hand in various ways, and the method I would use is the Newton-Raphson method.

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