Question

How do you write a polynomial function of minimum degree with real coefficients whose zeros include those listed: 5i and √5?

Answer 1

`f(x) = (x^2+25)(x-sqrt(5)) = x^3-sqrt(5)x^2+25-25sqrt(5)`

or if you want rational coefficients:

`g(x) = (x^2+25)(x^2-5) = x^4+20x^2-125`

Explanation:

If a polynomial has Real coefficients, then any non-Real Complex roots will occur in Complex conjugate pairs. So the roots of our polynomial must include `5i`, `-5i` and `sqrt(5)`.

If we allow irrational coefficients then the monic polynomial of lowest degree with these roots is:

`f(x) = (x-5i)(x+5i)(x-sqrt(5)) = (x^2+25)(x-sqrt(5))`

`= x^3-sqrt(5)x^2+25-25sqrt(5)`

To have rational coefficient then we also need the irrational conjugate `-sqrt(5)` of `sqrt(5)` resulting in:

`g(x) = (x-5i)(x+5i)(x-sqrt(5))(x+sqrt(5)) = (x^2+25)(x^2-5)`

`= x^4+20x^2-125`

Any polynomial with these roots will be a multiple (scalar or polynomial) of `f(x)` or `g(x)`