If a polynomial function with rational coefficients has the zeros #-3+sqrt5#, -i, what are the additional zeros?

1 Answer
marfre
Mar 3, 2017

#-3 - sqrt(5)# and #+i#

Explanation:

Complex zeros always come in pairs:

#x = -3 +-sqrt(5)# and #x = +-i#

The pairs form a quadratic equation:

  1. #(x+3-sqrt(5))(x+3+sqrt(5)) = # #x^2 + 3x +sqrt(5)x + 3x + 3sqrt(5) - sqrt(5)x -3 sqrt(5) - sqrt(5) sqrt(5) =# #x^2+ 6x - 5#

  2. #(x-i)(x+i) = x^2 + x i -x i -i^2 = x^2 +1#,
    Note: #i^2 = -1#

In summary: The additional zeros are #-3 - sqrt(5)# and #+i#

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