Find a polynomial with integer coefficients that satisfies the given conditions. \( P \) has degree 3 and zeros 3 and \( i \).

1. Since \( P(x) \) has integer coefficients and one of its zeros is \( i \), its complex conjugate \( -i \) is also a zero. 2. Therefore, the three zeros of the degree 3 polynomial are \( 3 \), \( i \), and \( -i \). 3. The polynomial can be written in factored form as: \[ P(x) = k(x-3)(x-i)(x+i) \] where \( k \) is a nonzero constant. To ensure integer coefficients, we can choose \( k=1 \). 4. Note that: \[ (x-i)(x+i)=x^2+1 \] 5. Substituting back, we have: \[ P(x) = (x-3)(x^2+1) \] 6. Expanding the polynomial: \[ P(x) = x(x^2+1)-3(x^2+1)=x^3+x-3x^2-3 \] Thus, rearranging the terms: \[ P(x)=x^3-3x^2+x-3 \] The polynomial with integer coefficients that satisfies the conditions is: \[ x^3-3x^2+x-3. \]