R has degree 4 and zeros 4 − 3i and 2, with 2 a zero of multiplicity 2

Question

R has degree 4 and zeros 4 − 3i and 2, with 2 a zero of multiplicity 2 

 

R(x)=

Answer 1

Assuming real coefficients, 4 + 3i must also be a zero.

 

So it is (x- 4 + 3i)(x - 4 - 3i)(x-2)(x-2)

 

          ((x-4)+3i)((x-4)-3i)(x-2)(x-2) 

 

         ((x-4)^2 +9)(x^2 - 4x + 4)

 

         (x^2 - 8x + 16 + 9)(x^2 - 4x + 4) 

 

         (x^2 - 8x + 25)(x^2 - 4x + 4)

 

        x^4 - 8x^3 + 25x^2 

               -8x^3  + 32x^2  - 100x

                           +4x^2   - 32x     + 100

 

    

       x^4 - 16x^3 + 61x^2 - 132x + 100 = R(x)

 

 

 

        

 

Answer 2

R(x) = (x - (4 + 3i))(x - (4 - 3i))(x - 2)²

 

R(x) = (x² - x(4 - 3i) - x(4 + 3i) + (16 - 9i²))(x - 2)²

 

R(x) = (x² - 8x  + 25)(x² - 4x + 4)