SOLUTION: find a third-degree polynomial equation with rational coefficients that has 1 and 3i as roots.

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Question 571790: find a third-degree polynomial equation with rational coefficients that has 1 and 3i as roots.
Answer by KMST(893) About Me  (Show Source):
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If the coefficients are rational numbers, irrational roots have to appear in conjugate pairs, so -3i must also be a root.
A polynomial of degree 3, with 1, 3i, and -3i as roots has to be equal to
K%28x-1%29%28x-3i%29%28x%2B3i%29 with some rational non-zero number K, and all polynomials of such form are of degree 3, and have 1, 3i, and -3i as roots.
So there are infinite such polynomials for the answer, but the simplest, with K=1 is
%28x-1%29%28x-3i%29%28x%2B3i%29=%28x-1%29%28x%5E2%2B9%29=x%5E3-x%5E2%2B9x-9