College Algebra, Chapter 4, Chapter Review, Section Review, Problem 48

Determine the polynomial of degree $4$ that has integer coefficients and zeros $3i$ and $4$, with $4$ a double zero.

Recall that if the polynomial function has real coefficients and if $a + bi$ is a zero of $P$, then $a - bi$ is also a zero of $P$. In our case the zeros are $3i, -3i$ and $4$ (multiplicity of $2$).

Thus, the required polynomial has the form


$
\begin{equation}
\begin{aligned}

P(x) =& (x - 3i) [x - (-3i)] (x - 4)^2
&&
\\
\\
=& (x - 3i)(x + 3i) (x - 4)^2
&&
\\
\\
=& (x^2 - 9i^2)(x - 4)^2
&& \text{Difference of squares}
\\
\\
=& (x^2 + 9)(x - 4)^2
&& \text{Recall that } i^2 = -1
\\
\\
=& (x^2 + 9) [x^2 - 8x + 16]
&& \text{Apply FOIL method}
\\
\\
=& x^4 - 8x^3 + 16x^2 + 9x^2 - 72x + 144
&& \text{Expand}
\\
\\
=& x^4 -8x^3 + 25x^2 - 72x + 144
&&

\end{aligned}
\end{equation}
$