Write the polynomial \( f(x) \) that meets the given conditions. Answers may vary. Degree 2 polynomial with zeros of \( 5+6 i \) and \( 5-6 i \). \( f(x)= \)

To find the polynomial \( f(x) \) with the given zeros \( 5 + 6i \) and \( 5 - 6i \), we can use the fact that a polynomial can be expressed in factored form based on its roots. The polynomial can be written as: \[ f(x) = (x - (5 + 6i))(x - (5 - 6i)) \] Now, let's expand this expression. First, we notice that \( (x - (5 + 6i))(x - (5 - 6i)) \) is a difference of squares, so we can calculate it as follows: \[ f(x) = (x - 5 - 6i)(x - 5 + 6i) = ((x - 5)^2 - (6i)^2) \] Since \( (6i)^2 = -36 \), we have: \[ f(x) = (x - 5)^2 + 36 \] Now, let's expand \( (x - 5)^2 \): \[ (x - 5)^2 = x^2 - 10x + 25 \] Putting it all together: \[ f(x) = x^2 - 10x + 25 + 36 = x^2 - 10x + 61 \] Therefore, the final polynomial is: \[ f(x) = x^2 - 10x + 61 \]