Question

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

Answer 1

Please see the explanation section below.

Explanation:

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`