Find a third-degree polynomial equation with rational coefficients that has the given numbers as roots. -5 and 1 - i

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Answer 1

if1-i is a root, so its complex conjugate, $1 + i$ . Write factors $(x - k)$ where k is a root for all 3 roots. Then multiply together.
$(x - (- 5)) (x - (1 - i)) (x - (1 + i)) = (x + 5) (x^{2} - x (1 + i) - x (1 - i) + 1 (1 - i) (1 + i))$
$= (x + 5) (x^{2} - x - i x - x + i x + 1 i i^{2})$
$= (x + 5) (x^{2} - 2 x + 2)$
$= x^{3} - 2 x^{2} + 2 x + 5 x^{2} - 10 x + 10$
$= x^{3} + 3 x^{2} - 8 x + 10$

Find a third-degree polynomial equation with rational coefficients that has the given numbers as roots. -5 and 1 - i

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