How do you write the simplest polynomial of least degree with zeros i, and 2i?

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How do you write the simplest polynomial of least degree with zeros i, and 2i?

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Answer 1 · 2017-05-24T11:19:57

Depending on what is meant by "simplest", $x^{2} - 3 i x - 2$ $x^2-3ix-2$ or $x^{4} + 5 x^{2} + 4$ $x^4+5x^2+4$

Explanation:

For the least degree polynomial (with complex coefficients) use

$(x - i) (x - 2 i) = x^{2} - 3 i x - 2$ $(x-i)(x-2i) = x^2-3ix-2$ .

If we need real coefficients, then, since $i$ $i$ is a zero, its conjugate $- i$ $-i$ must also be a zero. And similarly for $2 i$ $2i$ and $- 2 i$ $-2i$ .
In this case the polynomial is

$(x - i) (x + i) (x - 2 i) (x + 2 i) = (x^{2} + 1) (x^{2} + 4) = x^{4} + 5 x^{2} + 4$ $(x-i)(x+i)(x-2i)(x+2i) = (x^2+1)(x^2+4) = x^4+5x^2+4$

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How do you write the simplest polynomial of least degree with zeros i, and 2i?

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