College Algebra, Chapter 4, 4.4, Section 4.4, Problem 22

Determine all rational zeros of the polynomial $P(x) = x^3 - 4x^2 - 7x + 10$, and write the polynomial in factored form.

The leading coefficient of $P$ is $1$, so all the rational zeros are integers:

They are divisors of the constant term $10$. Thus, the possible candidates are

$\pm 1, \pm 2, \pm 5, \pm 10$

Using Synthetic Division







We find that $2$ is not a zero but that $1$ is a zero and that $P$ factors as

$x^3 - 4x^2 - 7x + 10 = (x - 1)(x^2 - 3x - 10)$

We now factor $x^2 - 3x - 10$ using trial and error, so


$
\begin{equation}
\begin{aligned}

x^3 - 4x^2 - 7x + 10 =& (x - 1)(x - 5)(x + 2)

\end{aligned}
\end{equation}
$


Therefore, the zeros of $P$ are $1, 5$ and $-2$.