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A degree 4 polynomial with integer coefficients has zeros −1−2i and 1, with 1 a zero of multiplicity 2. If the coefficient of x^4 is 1,
then the polynomial is

Mar 12, 2019
edited by Guest  Mar 12, 2019

#1
+7531
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$$f(x) = (x - 1)^2 (x - (-1-2i))(x - r) \text{ for some } r \in \mathbb C$$

But the polynomial has integer coefficients, so $$r = \overline{-1-2i} = -1+2i$$.

$$f(x) = (x^2 - 2x + 1) (x^2 + 2x + 5) \\f(x)= x^4+2x^2-8x+5$$

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Mar 12, 2019

#1
+7531
+1
$$f(x) = (x - 1)^2 (x - (-1-2i))(x - r) \text{ for some } r \in \mathbb C$$
But the polynomial has integer coefficients, so $$r = \overline{-1-2i} = -1+2i$$.
$$f(x) = (x^2 - 2x + 1) (x^2 + 2x + 5) \\f(x)= x^4+2x^2-8x+5$$