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# How do you write a polynomial function of least degree given the zeros -2i, 2+2sqrt2?

Mar 5, 2018

$a {x}^{2} - 2 a x \left(\sqrt{2} - i\right) - 4 i a \sqrt{2}$

#### Explanation:

If the roots to a polynomial are $\alpha$ and $\beta$

Then:

$\left(x - \alpha\right) \left(x - \beta\right)$ are factors of the polynomial.

So:

$a \left(x - \left(- 2 i\right)\right) \left(x - 2 \sqrt{2}\right)$

Where $a$ is a multiplier.

$a \left({x}^{2} - 2 x \sqrt{2} + 2 i x - 4 i \sqrt{2}\right)$

$a {x}^{2} - 2 a x \left(\sqrt{2} - i\right) - 4 i a \sqrt{2}$

This has a complex coefficient, since only one imaginary root was specified. To have real coefficients the roots must be complex conjugates.