Write a polynomial \( f(x) \) that satisfies the given conditions. Express the polynomial with the lowest possible leading positive integer coefficient. Polynomial of lowest degree with lowest possible integer coefficients, and with zeros \( 9-8 i \) and 0 (multiplicity 4). \( f(x)=\square \)

To construct the polynomial \( f(x) \) with the given roots, we first recognize that the root \( 9 - 8i \) has a complex conjugate, \( 9 + 8i \), which must also be included as a root. Therefore, the roots of \( f(x) \) are \( 9 - 8i \), \( 9 + 8i \), and \( 0 \) (with multiplicity 4). The polynomial can be expressed as: \[ f(x) = k(x)(x - (9 - 8i))(x - (9 + 8i)) \] where \( k \) is a constant. Let's first compute the product of the two complex factors: \[ (x - (9 - 8i))(x - (9 + 8i)) = (x - 9 + 8i)(x - 9 - 8i) = ((x - 9) + 8i)((x - 9) - 8i) \] Using the difference of squares: \[ = (x - 9)^2 - (8i)^2 = (x - 9)^2 + 64 \] Now expanding \( (x - 9)^2 \): \[ = x^2 - 18x + 81 + 64 = x^2 - 18x + 145 \] Now, including the multiplicity of 0: \[ f(x) = x^4(x^2 - 18x + 145) \] Distributing \( x^4 \): \[ f(x) = x^6 - 18x^5 + 145x^4 \] Finally, since the leading coefficient is already positive and the integer coefficients are the lowest possible, we have: \[ \boxed{x^6 - 18x^5 + 145x^4} \]