We are tasked with finding the zeros of the cubic polynomial function:

$P(x) = x^3 + 2x^2 - x - 2$

Step 1: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial must be a factor of the constant term (here, $-2$) divided by a factor of the leading coefficient (here, $1$).

Factors of $-2$: $\pm 1, \pm 2$

Thus, possible rational roots are: $\pm 1, \pm 2$.

Step 2: Test possible roots

Let's substitute the possible values into the polynomial to see if they yield zero.

• Test $x = 1$:

$P(1) = 1^3 + 2(1)^2 - 1 - 2 = 1 + 2 - 1 - 2 = 0$

Since $P(1) = 0$, $x = 1$ is a root.

Step 3: Perform synthetic division

Now that we know $x = 1$ is a root, we can perform synthetic division to factor the cubic polynomial.

Divide $P(x)$ by $(x - 1)$: