Finding a polynomial with integer coefficients that satisfy the given conditions

Question

What is the polynomial of R when R has degree 4 and zeros 1 − 5i and 3, with 3 a zero of multiplicity 2?

Answer 1

x=3

x-3=0

x-3 is one factor

with multiplity 2,

x-3 is a 2nd factor

or (x-3)^2

x=1-5i

x-1 =-5i

x^2-2x+1 =-25

x^2-2x+26 =0

x^2-2x+26 is another factor

multiply the factors together

R(x) = (x-3)^2(x^2-2x+26)

=(x^2-6x+9)(x^2-2x+26)

= x^4 -8x^3+23x^2+164x+234 is the 4th degree function with the given zeros and multiplity