I need some help on three homework questions, thank you!

1. Let P(x) = 0 be the polynomial equation of least possible degree, with rational coefficients, having \(\sqrt[3]{7} + \sqrt[3]{49}\) as a root. Compute the product of all of the roots of P(x) = 0.

2. A polynomial with integer coefficients is of the form \(3x^3 + a_2 x^2 + a_1 x - 6 = 0\).Enter all the possible integer roots of this polynomial, separated by commas.

3. The polynomial f(x) is divided by the polynomial d(x) to give a quotient of q(x) and a remainder of r(x). If deg f = 9 and deg r = 3, what is the maximum possible value of deg q?

Guest Apr 18, 2019

#1**+2 **

\(\quad (\sqrt[3]{7}+\sqrt[3]{49})^3\\ = 7 + 49 + 3\sqrt[3]{7\cdot49}(\sqrt[3]{7}+\sqrt[3]{49})\\ = 56 + 21(\sqrt[3]{7}+\sqrt[3]{49})\)

Therefore P(x) is x^{3} - 21x - 56.

Product of roots = \(-\dfrac{-56}{1} = 56\)

MaxWong Apr 18, 2019