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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:

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

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1 Answer 1

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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!)

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Eh. Difference of squares. I'm a fourth-year student in mathematics. Thank you! –  user23211 Jan 27 '12 at 16:40

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