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I can't solve the following questions: Let $a,b$ be real numbers, $Z= a + ib$.

  1. How much polynomials with complex coefficients $q(x) = x^3 + b_2 x^2 + b_1 x + b_0$ there are so that $Z$ is a root of this polynomial exactly twice?
  2. Find the coefficients of this polynomial so that $Z$ is a root of it exactly twice and the conjugate of $Z$ is also a root.

Thanks a lot for help!

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How do you know that you can't solve these questions? What have you tried (that didn't work, presumably)? Where did you get stuck? Please edit your question and include your thoughts on this. –  The Chaz 2.0 Oct 23 '13 at 17:16
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2 Answers 2

up vote 1 down vote accepted

Hint:$$p(x)=(x-Z)^2(x-Z^*)=x^3+b_2x^2+b_1x+b_0$$

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Thanks alot i used it to solve the second question, but about the first one. I understand that i can write each polynomial as (x-z1)(x-z2)... so if i got a polynomial with degree of 3, and a root z(twice) so it is: (x-z)^2 * (x-z1) so there can only be one z1 right? so the answer is only one polynomial? –  Michael Oct 23 '13 at 18:43
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$\textbf{Hint:}$ The polynomial with roots $z_1,z_2,z_3$ can be written as $(x-z_1)(x-z_2)(x-z_3)$.

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