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To find the real zeros of the function \( f(x) = x^3 - 4x^2 - 19x - 14 \) and use them to factor it, we’ll go through these steps:

### Step 1: Find Potential Rational Zeros
Using the **Rational Root Theorem**, potential rational zeros are factors of the constant term divided by factors of the leading coefficient. In this case:
- The constant term is \(-14\) (factors: \(\pm 1, \pm 2, \pm 7, \pm 14\))
- The leading coefficient is \(1\) (factors: \(\pm 1\))

This gives potential rational roots of \(\pm 1, \pm 2, \pm 7, \pm 14\).

### Step 2: Test Potential Zeros
We substitute each possible zero into \( f(x) \) to see if it equals zero.

#### Testing \( x = -1 \):
\[
f(-1) = (-1)^3 - 4(-1)^2 - 19(-1) - 14 = -1 - 4 + 19 - 14 = 0
\]
Since \( f(-1) = 0 \), \( x = -1 \) is a root of \( f(x) \).

### Step 3: Divide \( f(x) \) by \( x + 1 \)
Now, we use synthetic division to divide \( f(x) \) by \( x + 1 \).

1. **Set up synthetic division**:
   - Coefficients of \( f(x) \): \( 1, -4, -19, -14 \)
   - Zero \( x = -1 \)

2. **Synthetic division process**:
   ```
   -1 |  1  -4  -19  -14
       |    -1   5    14
       -----------------
         1  -5  -14   0
   ```
   - Quotient: \( x^2 - 5x - 14 \)
   - Remainder: \( 0 \)

This means \( f(x) = (x + 1)(x^2 - 5x - 14) \).

### Step 4: Factor \( x^2 - 5x - 14 \)
To further factor \( x^2 - 5x - 14 \), we can use the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-14)}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2} = \frac{5 \pm 9}{2}
\]
This gives:
\[
x = \frac{5 + 9}{2} = 7 \quad \text{and} \quad x = \frac{5 - 9}{2} = -2
\]

### Step 5: Write the Factorized Form
So, the real zeros are \( x = -1, 7, -2 \), and the factorized form of \( f(x) \) is:
\[
f(x) = (x + 1)(x - 7)(x + 2)
\]

### Final Answer
- **Real zeros of \( f(x) \)**: \( -1, 7, -2 \)
- **Factorized form of \( f(x) \)**: \( (x + 1)(x - 7)(x + 2) \)

Would you like more details on any step or have additional questions?

### Further Questions:
1. How would you verify if all roots are real using the discriminant?
2. Can you try synthetic division with a different potential zero?
3. What other factoring methods could we use if the polynomial was higher degree?
4. How does the Rational Root Theorem apply to other types of polynomials?
5. What is the importance of synthetic division in finding polynomial roots?

#### Tip:
For higher-degree polynomials, synthetic division can help break down complex expressions efficiently!

Find the real zeros of f(x) = x^3 - 4x^2 - 19x - 14 and use them to factor f.

Learn how to find the real zeros of the polynomial f(x) = x^3 - 4x^2 - 19x - 14 and use synthetic division and factoring to express it in factored form. This problem covers key algebraic concepts such as the Rational Root Theorem, synthetic division, and the quadratic formula. Suitable for high school students in Grades 9-12.

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