Create a monic polynomial with integer coefficients with $\sqrt[3]{2}+\sqrt[3]{3}+\sqrt[3]{5}$ as a root.
I want to create such polynomial but all the ways I have tried don't give a polynomial with integer coefficients and all of them give me irrational or non integer rationals. How should I work?
Please use high school math for solution and don't use matrix.
Best Answer
The systematic approach would be to write $x=\sqrt[3]{2}+\sqrt[3]{3}+\sqrt[3]{5}$ as a system of algebraic equations:
$$ \begin{align} \begin{cases} x - a - b - c & = 0 \\ a^3 - 2 & = 0 \\ b^3 - 3 & = 0 \\ c^3 - 5 & = 0 \end{cases} \end{align} $$
Eliminating $a,b,c$ between the equations will give a single equation in $x$ with integer coefficients.
Elimination can be done by repeatedly using the resultant of two polynomials, which is a routine calculation, thogh not pretty to do by hand in this particular case. Wolfram Alpha
resultant[ resultant[ resultant[ x - a - b - c, a^3 - 2, a], b^3 - 3, b ], c^3 - 5, c]gives:$$-x^{27} + 90 x^{24} - 1089 x^{21} + 62130 x^{18} - 105507 x^{15} + 16537410 x^{12} + 30081453 x^{9} + 1886601330 x^{6} - 73062900 x^{3} + 6859000$$