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

##### 1 Answer

#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

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

#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