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

$f(x) = 2x^3 + x^2 - 12x + 9$

To solve this, we can follow these steps:

Step 1: Apply the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of a polynomial is of the form $\frac{p}{q}$, where $p$ is a factor of the constant term (9), and $q$ is a factor of the leading coefficient (2).

Factors of 9: $\pm 1, \pm 3, \pm 9$
Factors of 2: $\pm 1, \pm 2$

Possible rational roots: $\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 9, \pm \frac{9}{2}$

Step 2: Test Possible Roots via Substitution

We begin by substituting possible values into $f(x)$.

Test $x = 1$:

$f(1) = 2(1)^3 + (1)^2 - 12(1) + 9 = 2 + 1 - 12 + 9 = 0$ Since $f(1) = 0$, $x = 1$ is a root.

Step 3: Factor the Polynomial

Since $x = 1$ is a root, we can factor $(x - 1)$ out of the cubic polynomial. Perform synthetic division to divide $2x^3 + x^2 - 12x + 9$ by $(x - 1)$.

Synthetic Division: