Answer:
The required polynomial is
.
Step-by-step explanation:
If a polynomial has degree n and
are zeroes of the polynomial, then the polynomial is defined as
![P(x)=a(x-c_1)(x-c_2)...(x-x_n)](https://tex.z-dn.net/?f=P%28x%29%3Da%28x-c_1%29%28x-c_2%29...%28x-x_n%29)
It is given that the polynomial R has degree 4 and zeros 3 − 3i and 2. The multiplicity of zero 2 is 2.
According to complex conjugate theorem, if a+ib is zero of a polynomial, then its conjugate a-ib is also a zero of that polynomial.
Since 3-3i is zero, therefore 3+3i is also a zero.
Total zeroes of the polynomial are 4, i.e., 3-3i, 3_3i, 2,2. Let a=1, So, the required polynomial is
![R(x)=(x-3+3i)(x-3-3i)(x-2)(x-2)](https://tex.z-dn.net/?f=R%28x%29%3D%28x-3%2B3i%29%28x-3-3i%29%28x-2%29%28x-2%29)
![R(x)=((x-3)+3i)((x-3)-3i)(x-2)^2](https://tex.z-dn.net/?f=R%28x%29%3D%28%28x-3%29%2B3i%29%28%28x-3%29-3i%29%28x-2%29%5E2)
![[a^2-b^2=(a-b)(a+b)]](https://tex.z-dn.net/?f=%5Ba%5E2-b%5E2%3D%28a-b%29%28a%2Bb%29%5D)
![R(x)=(x^2-6x+9-9(i)^2((x-3)-3i)(x-2)^2](https://tex.z-dn.net/?f=R%28x%29%3D%28x%5E2-6x%2B9-9%28i%29%5E2%28%28x-3%29-3i%29%28x-2%29%5E2)
![[i^2=-1]](https://tex.z-dn.net/?f=%5Bi%5E2%3D-1%5D)
![R(x)=(x^2-6x+18)(x^2-4x+4)](https://tex.z-dn.net/?f=R%28x%29%3D%28x%5E2-6x%2B18%29%28x%5E2-4x%2B4%29)
![R(x)=x^4-10x^3+46x^2-96x+72](https://tex.z-dn.net/?f=R%28x%29%3Dx%5E4-10x%5E3%2B46x%5E2-96x%2B72)
Therefore the required polynomial is
.