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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:

\[
\begin{array}{r|rrrr}
1 & 2 & 1 & -12 & 9 \\
  &   & 2 & 3 & -9 \\
\hline
  & 2 & 3 & -9 & 0 \\
\end{array}
\]

The quotient is \( 2x^2 + 3x - 9 \).

### Step 4: Solve the Quadratic Equation
We now solve the quadratic equation \( 2x^2 + 3x - 9 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 2 \), \( b = 3 \), and \( c = -9 \).

\[
x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-9)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 72}}{4} = \frac{-3 \pm \sqrt{81}}{4} = \frac{-3 \pm 9}{4}
\]

This gives two solutions:
\[
x = \frac{-3 + 9}{4} = \frac{6}{4} = \frac{3}{2}
\]
\[
x = \frac{-3 - 9}{4} = \frac{-12}{4} = -3
\]

### Step 5: List the Zeros
The zeros of \( f(x) \) are \( x = 1, \frac{3}{2}, -3 \).

### Final Answer:
The zeros of the polynomial, arranged from smallest to largest, are:

\[
x = -3, 1, \frac{3}{2}
\]

There are no double roots in this case.

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Let me know if you need further details or clarification!

Here are five related questions to expand understanding:
1. How can synthetic division be applied to factor higher-degree polynomials?
2. What is the Rational Root Theorem, and why is it useful?
3. How would you handle polynomials with complex roots?
4. Can all cubic polynomials be factored easily using rational roots?
5. What methods exist to approximate irrational roots of polynomials?

**Tip:** When using the quadratic formula, always carefully calculate the discriminant (\( b^2 - 4ac \)) to check if roots are real or complex.

Find all the zeros of f(x): f(x) = 2x^3 + x^2 - 12x + 9. Arrange your answers from smallest to largest. If there is a double root, list it twice.

Learn how to find all the zeros of the cubic polynomial f(x) = 2x³ + x² - 12x + 9, step-by-step using the Rational Root Theorem and Quadratic Formula. Suitable for high school students (Grades 9-12).

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