Let there be a polynomial function, with integer coefficients, of least degree such that $7^\frac{1}3 + 49^\frac13$ is a root of the polynomial. What is the product of the roots of the $f(x) = 0$?

In problems which state the degree of the polynomial as least degree (with additional relevant information), is it assumed that we have to take the polynomial as degree 3 or 2? Considering how tedious the cubic polynomial solution is, I don't think that is to be used almost anywhere. How is one supposed to go about such a problem?

  • 1
    $\begingroup$ Are you aware of Vieta's formulas? $\endgroup$ – Cataline Nov 22 '15 at 12:16
  • $\begingroup$ Yes I am @Cataline $\endgroup$ – Sat D Nov 22 '15 at 12:17
  • 1
    $\begingroup$ Try cubing the expression, and see what the result is. The minimum polynomial becomes clear after you have cubed it, i you are observant enough. $\endgroup$ – Cataline Nov 22 '15 at 12:20
  • $\begingroup$ Ah damn, should have seen that. Thanks! $\endgroup$ – Sat D Nov 22 '15 at 12:22
  • 2
    $\begingroup$ The least degree polynomial is $x-7^\frac{1}3 -49^\frac13$. $\endgroup$ – Bernard Nov 22 '15 at 12:37

Let $\alpha=7^{\frac13}$ so $\alpha^3=7$. We need to calculate the minimal polynomial of $x=\alpha+\alpha^2$. One has $x^3=\alpha^3+3\alpha^2\alpha^2+3\alpha\alpha^4+\alpha^6=7+3\cdot7\alpha+3\cdot7\alpha^2+49=56+21x$

Thus the searched minimal polynomial is $p(x)=x^3-21x-56$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.