To factor the polynomial \( f(x) = x^3 - 5x^2 + 2x + 8 \), we can start by using the Rational Root Theorem to test possible rational roots. The potential rational roots are the factors of the constant term (8) divided by the factors of the leading coefficient (1). This gives us the possible roots: \( \pm 1, \pm 2, \pm 4, \pm 8 \).

### Step 1: Testing Possible Roots

Let's test \( x = 1 \):

\[
f(1) = 1^3 - 5(1^2) + 2(1) + 8 = 1 - 5 + 2 + 8 = 6 \quad \text{(not a root)}
\]

Next, test \( x = -1 \):

\[
f(-1) = (-1)^3 - 5(-1)^2 + 2(-1) + 8 = -1 - 5 - 2 + 8 = 0 \quad \text{(is a root)}
\]

### Step 2: Synthetic Division

Since \( x = -1 \) is a root, we can use synthetic division to divide \( f(x) \) by \( x + 1 \).

Perform synthetic division with \( -1 \):

```
-1 |  1  -5   2   8
   |      -1   6  -8
   -------------------
     1  -6   8   0
```

This gives us:

\[
f(x) = (x + 1)(x^2 - 6x + 8)
\]

### Step 3: Factoring the Quadratic

Next, we need to factor \( x^2 - 6x + 8 \):

\[
x^2 - 6x + 8 = (x - 2)(x - 4)
\]

### Final Factored Form

Thus, the complete factorization of \( f(x) \) is:

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

### Summary

The factored form of \( f(x) \) is \( (x + 1)(x - 2)(x - 4) \).

If you need any further details or have questions, feel free to ask! Here are some follow-up questions:

1. Would you like to see the graph of the polynomial?
2. Do you want to learn about other methods for factoring polynomials?
3. Are you interested in how to apply these techniques to higher degree polynomials?
4. Would you like to explore the roots of the polynomial in more detail?
5. Do you need help with similar problems? 

**Tip:** Always check potential roots using synthetic division to simplify polynomials before attempting complex factorizations!

Explore how to factor the polynomial f(x) = x^3 - 5x^2 + 2x + 8 using the Rational Root Theorem and synthetic division. This step-by-step guide helps students understand polynomial factorization concepts. Suitable for Grades 9-12.

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