How do you find a third degree polynomial given roots $5$ and $2i$ ?

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Answer 1 · 2016-11-07T21:11:32

Please see the explanation section below.

For a third degree polynomial, we need 3 linear factors.

Since $5$ and $2i$ are roots (zeros), we know that $x-5$ and $x-2i$ are factors.

If we want a polynomial with real coeficients, then the complex conjugate of $2i$ (which is $-2i$ ) must also be a root and $x+2i$ must be a factor.

One polynomial with real coefficients that meets the requirements is

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

$= x^3-5x^2+4x-20$

Any constant multiple of this also meets the requirements.

For example

$7(x^3-5x^2+4x-20) = 7x^3-35x^2+28x-140$

How do you find a third degree polynomial given roots 55 and 2i2i?