A degree 4 polynomial P (x) with integer coefficients has zeros 5 i and 3, with 3 being a zero of multiplicity 2. Moreover, the coefficient of x^4 is 1. Find the polynomial.

Multiplicity is how many times the root repeats

roots r1 and r2 of a polynomial factor to

(x-r1) (x-r2)

so 3 multiplicty 2 means (x-3) ^2 is in the factorization of that polynomial

also, for a polynomial with real coefients, if a+bi is a roots, a-bi is also a root

5i is a root, therefor - 5i is a root as well

roots are

(x-5i) (x+5i) (x-3) ^2

if we expand

x^4-6x^3+34x^2-150x+225

the polynomial is

f (x) = x^4-6x^3+34x^2-150x+225