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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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It could also be a quadratic times a cubic. – Will Jagy Oct 21 '13 at 0:42
Over the reals, this factors as a linear term and two quadratics -- none of them at all nice. – vadim123 Oct 21 '13 at 0:52
@StefanSmith, if you ask Wolfram Alpha kindly to factor it for you, it will give you numerical answers, showing one real root and four complex roots which appear to be conjugate pairs. – dfeuer Oct 21 '13 at 1:05
@dfeuer I'm sorry but I can't use Wolfram Alpha during tests. That would be nice though! So do you think you could show me exactly what you did? – Jessica Oct 21 '13 at 1:13
To sketch the function you don't need to factor the polynomial; just plug in a few points. – vadim123 Oct 21 '13 at 1:24

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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how would this particular method give me an approximate root by hand for x^5 + 20x^2 + 5=0? – Jessica Oct 21 '13 at 1:33
@Jessica, why don't you read the article and try it yourself? It's not hard, just tedious. – dfeuer Oct 21 '13 at 1:41
@dfeuer ahaha now I dont want to solve this problem nearly as bad anymore :( I give up. – Jessica Oct 21 '13 at 2:53
@Jessica: If you are initially intimidated, you can use the freely downloadable software called Pari/GP to factor polynomials. You can also do it at one of the websites like Wolfram Alpha. But remember that for some it is more interesting to look for the theory behind this stuff than just finding an answer. – Doldrums Oct 23 '13 at 4:09

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