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# help polynomials

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I cant solve this

Let $P(x) = 0$ be the polynomial equation of least possible degree, with rational coefficients, having $\sqrt{2} + \sqrt{3}$ as a root.  Compute the product of all of the roots of $P(x) = 0.$

Aug 18, 2023

#1
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I think this problem is easier than it may first appear unless I am misreading the question.

First, I would consider the given information and determine if I can fit all the criteria in a first-degree polynomial. I think I can. Just do this: $$P(x) = x - \left(\sqrt{2} + \sqrt{3}\right)$$. This is a first-degree polynomial, so this must be of least degree, and P(x) has rational coefficients. The coefficients of the x-term is 1, which is rational. Because of the way I have written this polynomial, it is guaranteed to have the root of $$\sqrt{2} + \sqrt{3}$$.

There is only one root, so the product of the roots is just $$\sqrt{2} + \sqrt{3}$$.

Aug 19, 2023
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I did some research after posting this answer, and I am relatively certain that the answer I posted previously is not correct, but I will leave this here in case it helps someone else solve this problem. In a polynomial in the form $$P(x) = a_nx^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \cdots + a_2x^2 + a_1x + a_0$$, it seems like there is some ambiguity about whether or not $$a_0$$ fits the definition of a coefficient or not. I always considered $$a_0$$ as classified as a constant, but some argue that $$a_0$$ is a coefficient of the $$x^0$$ term. If that interpretation fits this problem, then the answer I posted previously is not correct. I made a few attempts, but I was unable to solve this problem with the new interpretation.

The3Mathketeers  Aug 19, 2023