College Algebra, Chapter 4, 4.3, Section 4.3, Problem 64

Determine a polynomial of degree $4$ that has integer coefficient and zeros $1, -1, 2$ and $\displaystyle \frac{1}{2}$.

By factor theorem $x - 1, x - (-1), x - 2$ and $\displaystyle x - \frac{1}{2}$ must all be factors of the desired polynomial, so let


$
\begin{equation}
\begin{aligned}

P(x) =& (x - 1)(x + 1) (x - 2)\left(x - \frac{1}{2} \right)
&& \text{Distributive Property}
\\
\\
P(x) =& (x^2 - 1) \left( x^2 - \frac{1}{2} x - 2x + 1 \right)
&& \text{Simplify}
\\
\\
P(x) =& (x^2 - 1) \left( x^2 - \frac{5}{2} x + 1 \right)
&& \text{Distributive Property}
\\
\\
P(x) =& x^4 - \frac{5}{2} x^3 + x^2 - x^2 + \frac{5}{2} x - 1
&& \text{Simplify}
\\
\\
P(x) =& x^4 - \frac{5}{2} x^3 + \frac{5}{2} x - 1
&& \text{Multiply 2, since it should have integer coefficient}
\\
\\
P(x) =& 2x^4 - 5x^3 + 5x - 2
&&



\end{aligned}
\end{equation}
$

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