Question

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.

Answer 1

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