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

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Answer 1 · 2018-03-05T11:26:15

$ax^2-2ax(sqrt(2)-i)-4iasqrt(2)$

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

Then:

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

So:

$a(x-(-2i))(x-2sqrt(2))$

Where $a$ is a multiplier.

$a(x^2-2xsqrt(2)+2ix-4isqrt(2))$

$ax^2-2ax(sqrt(2)-i)-4iasqrt(2)$

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

How do you write a polynomial function of least degree given the zeros -2i, 2+2sqrt22+2sqrt2?