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Let's find all rational zeros of the polynomial:

\[
P(x) = 12x^3 - 20x^2 + x + 3
\]

### Step 1: Use the Rational Root Theorem
The Rational Root Theorem helps us find potential rational zeros of a polynomial. The potential rational roots are of the form:

\[
\frac{p}{q}
\]

where:
- \( p \) is a factor of the constant term (in this case, 3).
- \( q \) is a factor of the leading coefficient (in this case, 12).

#### Factors of the constant term (\( p \)):
- The factors of 3 are \( \pm 1, \pm 3 \).

#### Factors of the leading coefficient (\( q \)):
- The factors of 12 are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

#### Potential Rational Roots:
The possible rational roots are:

\[
\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{6}, \pm \frac{3}{12}
\]

This can be simplified to:

\[
\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}
\]

### Step 2: Test the Potential Roots
Now, we substitute the potential rational roots into \( P(x) \) to determine which of them are actual roots. Let's test a few values:

1. **\( x = 1 \)**:

\[
P(1) = 12(1)^3 - 20(1)^2 + 1 + 3 = 12 - 20 + 1 + 3 = -4
\]

Not a root.

2. **\( x = -1 \)**:

\[
P(-1) = 12(-1)^3 - 20(-1)^2 + (-1) + 3 = -12 - 20 - 1 + 3 = -30
\]

Not a root.

3. **\( x = 3 \)**:

\[
P(3) = 12(3)^3 - 20(3)^2 + 3 + 3 = 12 \times 27 - 20 \times 9 + 6 = 324 - 180 + 6 = 150
\]

Not a root.

4. **\( x = -3 \)**:

\[
P(-3) = 12(-3)^3 - 20(-3)^2 + (-3) + 3 = -324 - 180 - 3 + 3 = -504
\]

Not a root.

5. **\( x = \frac{1}{2} \)**:

\[
P\left(\frac{1}{2}\right) = 12 \left(\frac{1}{2}\right)^3 - 20 \left(\frac{1}{2}\right)^2 + \frac{1}{2} + 3 = 12 \times \frac{1}{8} - 20 \times \frac{1}{4} + \frac{1}{2} + 3
\]

\[
= \frac{12}{8} - \frac{20}{4} + \frac{1}{2} + 3 = 1.5 - 5 + 0.5 + 3 = 0
\]

**\( x = \frac{1}{2} \)** is a root.

### Step 3: Factor the Polynomial Using Synthetic Division
Since \( x = \frac{1}{2} \) is a root, we can factor \( P(x) \) using synthetic division.

Using synthetic division with \( x - \frac{1}{2} \):

\[
\begin{array}{r|rrrr}
\frac{1}{2} & 12 & -20 & 1 & 3 \\
            &    & 6  & -7 & -3.5 \\
\hline
            & 12 & -14 & -6 & -

Find all rational zeros of the polynomial P(x) = 12x^3 - 20x^2 + x + 3. Enter your answers as a comma-separated list, including repetitions.

Learn how to find all rational zeros of the polynomial P(x) = 12x^3 - 20x^2 + x + 3 using the Rational Root Theorem and synthetic division. This guide provides a step-by-step solution suitable for high school students in grades 9-12.

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