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Is there any algorithm that can be used to find all the possible roots of a polynomial?

For example, I'd like to find all possible roots of the polynomial $x^3 + 3x^2 + 2x + 6$.

If I remember correctly, the possible rational roots of a polynomial are given by all factors of the constant term, divided by all factors of the leading coefficient - is this true, or is it false?

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    http://en.wikipedia.org/wiki/Rational_root_theorem2012-11-04
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    @TheChaz Now is there an efficient algorithm for finding the irrational roots of a polynomial?2012-11-04
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    Yes --- Newton's Method, which see.2012-11-04
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    dichotomy, newton, etc...2012-11-04
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    @F'OlaYinka Can you explain what "dichotomy" refers to in this context?2012-11-04
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    @AndersonGreen http://www.encyclopediaofmath.org/index.php/Dichotomy_method2012-11-04
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    This particular polynomial is easily factored: $x^3+3x^2+2x+6$ $=x^2(x+3)+2(x+3)$ $=(x^2+2)(x+3)$ $=(x+i\sqrt{2})(x-i\sqrt{2})(x+3)$. That's equal to $0$ if and only if one of the three factors is equal to $0$. So, for example, one of the roots is $-3$.2012-11-04

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