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

To find a polynomial \( P \) with integer coefficients, we start by noting that if \( i \) is a zero, then its conjugate \( -i \) must also be a zero (since the coefficients are integers). Thus, the zeros of the polynomial are \( 3 \), \( i \), and \( -i \). The polynomial can be expressed as: \[ P(x) = (x - 3)(x - i)(x + i) \] We can simplify the product of the complex conjugate factors: \[ (x - i)(x + i) = x^2 + 1 \] Now we can substitute this back into the polynomial: \[ P(x) = (x - 3)(x^2 + 1) \] Next, we distribute \( (x - 3) \) into \( (x^2 + 1) \): \[ P(x) = x(x^2 + 1) - 3(x^2 + 1) = x^3 + x - 3x^2 - 3 = x^3 - 3x^2 + x - 3 \] Thus, the polynomial \( P(x) \) is: \[ P(x) = x^3 - 3x^2 + x - 3 \]