I cant solve this

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

maximum Aug 18, 2023

#1**+1 **

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: \(P(x) = x - \left(\sqrt[3]{2} + \sqrt[3]{3}\right)\). 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 \(\sqrt[3]{2} + \sqrt[3]{3}\).

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

The3Mathketeers Aug 19, 2023

#2**0 **

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