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 \( 8-3 i \) and 0 (multiplicity 5). \( f(x)=\square \)

To find the polynomial \( f(x) \) with the given zeros \( 8 - 3i \) and \( 0 \) (with multiplicity 5), we first need to include the conjugate root, \( 8 + 3i \), since the coefficients must be real. The zeros are: 1. \( 0 \) (with multiplicity 5) 2. \( 8 - 3i \) 3. \( 8 + 3i \) The polynomial can be constructed as follows: 1. From the zero \( 0 \) with multiplicity 5, we have the factor \( x^5 \). 2. From the zero \( 8 - 3i \) and its conjugate \( 8 + 3i \), we have the factors \( (x - (8 - 3i))(x - (8 + 3i)) \). This simplifies to \( (x - 8 + 3i)(x - 8 - 3i) \). Calculating the product of the complex factors: \[ (x - 8 + 3i)(x - 8 - 3i) = ((x - 8)^2 - (3i)^2) = (x - 8)^2 + 9 \] Expanding it: \[ = (x^2 - 16x + 64 + 9) = x^2 - 16x + 73 \] Now, we put everything together: \[ f(x) = x^5(x^2 - 16x + 73) \] To express it in standard polynomial form: \[ f(x) = x^7 - 16x^6 + 73x^5 \] Thus, the polynomial is: \[ f(x) = x^7 - 16x^6 + 73x^5 \]