2
$\begingroup$

This is a problem from a test in my course in analytic functions. I didn't manage to solve it. Could you please give me a hint? The problem is:

Calculate the third root of the sum of the coefficients of the polynomial whose roots are the squares of the roots of the following polynomial:

$P(z)=z^5-z+11.$

1 Answers 1

0

Let's do this for a polynomial of degree $2$, with omission of some steps that you will probably be able to fill in yourself: $ P(z) = (z-a)(z-b), $ then the polynomial under investigation is $ Q(z) = (z-a^2)(z-b^2) $ The sum of its coefficients is $Q(1)$: $ Q(1) = (1-a)(1+a)(1-b)(1+b) = P(1)P(-1) $

So the answer would be $ \sqrt[3]{ P(1)P(-1)},$

I'll leave the details and generalisation to other degrees to you. (Note that there is an extra $-1$ appearing for odd degrees!)

  • 0
    Eh. Difference of squares. I'm a fourth-year student in mathematics. Thank you!2012-01-27