R has degree 4 and zeros 2 − 5i and 2, with 2 a zero of multiplicity 2.

Answer 1

Assuming real coefficients, 2 + 5i must also be a zero.

 

So it is (x- 2 + 5i)(x - 2 - 5i)(x-2)(x-2) = R(x)

 

     ((x-2)+5i)((x-2)-5i)(x-2)(x-2) = R(x)

 

     ((x-2)^2+25)(x^2-4x+4) = R(x)

(x^2-4x+4+25)(x^2-4x+4) = R(x)

(x^2-4x+29)(x^2-4x+4) = R(x)

x^4-4x^3+4x^2-4x^3-16x^2-16x+29x^2-116x+116 = R(x)

x^4-8x^3+17x^2-132x+116 = R(x)