Write function from roots

Question

x = -2 ± 5i

 

and

 

 

One root is 6i

Answer 1

A function with the roots -2 + 5i and -2 - 5i is:

 

  f(x) = [x - (-2+5i)][x - (-2-5i)]

 

          = [(x-2) - 5i][(x-2) + 5i]

 

          = (x-2)² - (5i)²  = x² - 4x + 4 - 25i²

 

                                       = x² - 4x + 4 + 25  = x² - 4x + 29

 

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If a polynomial function, f(x), has real coefficients and if 6i is a root, then -6i must also be a root.  

 

So, a possible function would be f(x) = [x - 6i][x - (-6i)]

 

                                                                 = (x - 6i)(x + 6i)

 

                                                                 = x² - 36i²

 

                                                                 = x² + 36 

Answer 2

Are you saying that -2 ± 5i and 6i are all roots?  The other root is -6i.