+818 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 $P(x) = 0.$
+189 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−(3√2+3√3). 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 3√2+3√3.
There is only one root, so the product of the roots is just 3√2+3√3.
+189 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)=anxn+an−1xn−1+an−2xn−2+⋯+a2x2+a1x+a0, it seems like there is some ambiguity about whether or not a0 fits the definition of a coefficient or not. I always considered a0 as classified as a constant, but some argue that a0 is a coefficient of the x0 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.
