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Suppose you have a consistent radical function $R(x)$ that can be solved by finding the roots of an nth degree polynomial, assumed to be solvable using radicals. Some of the $n$ roots maybe extraneous. Typically the student is told to verify each possible root by inserting it into $R(x)$ and verifying that $R(x_i) = 0$ for $i= 1,..,n$. This step basically amounts to verifying an identity that can involve multiple roots and powers. And the $x_i$'s can be complex. Can it be proved algebraically that each extraneous root can be identified, i.e., $R(x_i) \neq 0$ for some $i$?

Let me try to be specific. Suppose $R(x) = \sqrt{x+a} -(x+b) = 0$. The solution to this simple radical equation results in a quadratic equation. When two roots result it appears that one of them will be extraneous. The roots can be complex involving irrational values. Verifying that a root of the polynomial is also a root of $R(x)$ requires proving an identity: $R(x_i)=0$. This could be very difficult because the root may involve nested radicals and $R(x)$ can result in more nested radicals. But how does one prove the identity without resorting to steps that were used to generate the polynomial? I hope this helps. My gut feeling is that one cannot be assured of proving the identity.

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    Could you please give a concrete example of what you have in mind?2011-06-02
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    Related: http://math.stackexchange.com/questions/39944/is-there-an-algorithm-to-find-the-roots-of-high-order-polynomials/42687#426872011-06-02
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    I think this may be a clearer reformulation of the question: given a function $R(x)$ that may involve radicals and the usual arithmetic operations, and a number $\alpha$ expressed in terms of radicals and the usual arithmetic operations, is there an algorithm guaranteed to tell you whether or not $R(\alpha)=0$?2011-06-02

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