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

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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:
$$
x^3-3x^2+x-3.
$$
The polynomial with integer coefficients that has zeros 3 and $$ i $$ is: $$ x^3 - 3x^2 + x - 3 $$

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

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