College Algebra, Chapter 4, 4.5, Section 4.5, Problem 38

Find a polynomial $P(x)$ of degree 3 that has integer coefficients and zeros and $i$.
Recall that if the polynomial function $P$ has real coefficient and if $a + bi$ is a zero of $P$, then $a - bi$ is also a zero of $P$. In our case, we have zeros of $0, i$ and $-i$. Thus

$
\begin{equation}
\begin{aligned}
P(x) &= (x-0)(x-i)(x+i) && \text{Model}\\
\\
&= x(x^2-i^2) && \text{Difference of squares}\\
\\
&= x(x^2+1) && \text{Recall that } i^2 = -1\\
\\
&= x^3 + x && \text{Simplify}
\end{aligned}
\end{equation}
$