Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros. \( -3 i, 6 \) \( \begin{array}{l}\text { The polynomial function in expanded form is } f(x)=\square \\ \text { (Use } 1 \text { for the leading coefficient.) }\end{array} \)

To find a polynomial function of the lowest degree with rational coefficients and the given zeros \(-3i\) and \(6\), we need to remember that the complex zeros come in conjugate pairs. Thus, since \(-3i\) is one zero, its conjugate \(3i\) is also a zero. Now, the zeros of the polynomial are \(-3i\), \(3i\), and \(6\). The polynomial can be formed by deriving factors from these zeros: 1. For the zeros \( -3i \) and \( 3i \): \[ (x + 3i)(x - 3i) = x^2 + 9 \] (this is obtained using the difference of squares where \(a^2 + b^2 = (x^2 + 9)\)). 2. For the zero \( 6 \): \[ (x - 6) \] Now, combine these factors to form the polynomial: \[ f(x) = (x^2 + 9)(x - 6) \] Next, we expand this expression: \[ f(x) = x^3 - 6x^2 + 9x - 54 \] So, the polynomial function in expanded form is: \[ f(x) = x^3 - 6x^2 + 9x - 54 \]