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The given function is:

\[
f(x) = x^3 - x^2 - 11x + 3
\]

We need to find all the zeros of this cubic function. Let's solve for the roots (or zeros) by using methods such as factoring, synthetic division, or the Rational Root Theorem to narrow down potential rational roots.

I'll start solving this step by step.

### Step 1: Apply the Rational Root Theorem
The Rational Root Theorem suggests that any possible rational root is a factor of the constant term (3) divided by a factor of the leading coefficient (1 in this case).

The possible rational roots are:
\[
\pm 1, \pm 3
\]

### Step 2: Check possible roots
We can substitute these values into the equation to see which one is a root.

#### Try \( x = 1 \):
\[
f(1) = 1^3 - 1^2 - 11(1) + 3 = 1 - 1 - 11 + 3 = -8 \quad \text{(not a root)}
\]

#### Try \( x = -1 \):
\[
f(-1) = (-1)^3 - (-1)^2 - 11(-1) + 3 = -1 - 1 + 11 + 3 = 12 \quad \text{(not a root)}
\]

#### Try \( x = 3 \):
\[
f(3) = 3^3 - 3^2 - 11(3) + 3 = 27 - 9 - 33 + 3 = -12 \quad \text{(not a root)}
\]

#### Try \( x = -3 \):
\[
f(-3) = (-3)^3 - (-3)^2 - 11(-3) + 3 = -27 - 9 + 33 + 3 = 0 \quad \text{(root found!)}
\]

So, \( x = -3 \) is a root of the polynomial.

### Step 3: Divide the polynomial by \( x + 3 \)
We now divide the cubic polynomial \( x^3 - x^2 - 11x + 3 \) by \( x + 3 \) using synthetic division.

Let's perform synthetic division.

After performing synthetic division, we obtain the quotient:

\[
x^2 - 4x + 1
\]

The remainder is 0, confirming that \(x = -3\) is indeed a root, and the original cubic polynomial factors as:

\[
f(x) = (x + 3)(x^2 - 4x + 1)
\]

### Step 4: Solve the quadratic equation
Now, we solve the quadratic \(x^2 - 4x + 1 = 0\) using the quadratic formula:

\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(1)}}{2(1)}
\]

\[
x = \frac{4 \pm \sqrt{16 - 4}}{2}
\]

\[
x = \frac{4 \pm \sqrt{12}}{2}
\]

\[
x = \frac{4 \pm 2\sqrt{3}}{2}
\]

\[
x = 2 \pm \sqrt{3}
\]

### Step 5: Final Answer
The zeros of the function are:

\[
x = -3, \quad x = 2 + \sqrt{3}, \quad x = 2 - \sqrt{3}
\]

These are the exact values for the zeros.

Would you like more details on any step or further clarification?

Here are 5 related questions to expand your understanding:
1. How is the Rational Root Theorem used to find potential rational roots of a polynomial?
2. What is the process of synthetic division, and why is it useful?
3. How do you solve quadratic equations using the quadratic formula?
4. Can all cubic polynomials be factored easily? If not, what are other methods?
5. What are the benefits of finding exact values instead of decimal approximations?

**Tip:** Always try factoring or using the Rational Root Theorem for polynomials before moving to more advanced methods like numerical approximations.

Find all zeros of f(x) = x^3 - x^2 - 11x + 3. Enter the zeros separated by commas. Enter exact values, using radicals and/or fractions if necessary, not decimal approximations.

This page explains how to find all zeros of the cubic function f(x) = x^3 - x^2 - 11x + 3 using the Rational Root Theorem and synthetic division. It also covers how to use the quadratic formula to solve for the remaining roots after factoring.

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