https://file.aiquickdraw.com/mathbot/file/1725122212833own5zojt.jpg

Let's go through the problem step-by-step.

### Problem Analysis:

The given problem involves a polynomial \( p(x) = x^3 - 4x^2 + cx + 14 \) with \( c \) as a variable coefficient. The task involves solving various sub-questions related to this polynomial.

### Step-by-Step Solution:

1. **Finding the unknown coefficient \( c \) using a known root:**

   From the text, it appears that \( x = -2 \) is a root of the polynomial. This means that when \( x = -2 \), the polynomial \( p(-2) = 0 \).

   Substitute \( x = -2 \) into the polynomial:

   \[
   p(-2) = (-2)^3 - 4(-2)^2 + c(-2) + 14 = 0
   \]

   Simplify this equation:

   \[
   -8 - 16 - 2c + 14 = 0
   \]

   Combine like terms:

   \[
   -10 - 2c = 0
   \]

   Solving for \( c \):

   \[
   -2c = 10 \implies c = -5
   \]

2. **Verifying the complete factored form of the polynomial:**

   Now that \( c = -5 \), substitute it back into the polynomial:

   \[
   p(x) = x^3 - 4x^2 - 5x + 14
   \]

   To factorize \( p(x) \), we divide by the known factor \( (x + 2) \) using synthetic division:

   Perform synthetic division with \( x + 2 \):

   ```
   -2 |  1  -4  -5   14
        |     -2   12  -14
        -----------------
          1  -6    7    0
   ```

   The quotient is \( x^2 - 6x + 7 \).

   Thus, the factorized form is:

   \[
   p(x) = (x + 2)(x^2 - 6x + 7)
   \]

3. **Finding the remaining roots of the polynomial:**

   The quadratic part is \( x^2 - 6x + 7 = 0 \).

   Use the quadratic formula:

   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{6 \pm \sqrt{36 - 28}}{2} = \frac{6 \pm \sqrt{8}}{2} = \frac{6 \pm 2\sqrt{2}}{2} = 3 \pm \sqrt{2}
   \]

   The roots are \( x = 3 + \sqrt{2} \) and \( x = 3 - \sqrt{2} \).

4. **Sum of the numerical roots:**

   The numerical roots found are \( -2 \), \( 3 + \sqrt{2} \), and \( 3 - \sqrt{2} \).

   Their sum:

   \[
   -2 + (3 + \sqrt{2}) + (3 - \sqrt{2}) = -2 + 6 = 4
   \]

5. **Product of the numerical roots:**

   For a polynomial \( ax^3 + bx^2 + cx + d \), the product of the roots is given by \( -\frac{d}{a} \).

   Here:

   \[
   \text{Product} = -\frac{14}{1} = -14
   \]

### Summary:
- **Coefficient \( c \)**: \( -5 \)
- **Roots**: \( x = -2, 3 + \sqrt{2}, 3 - \sqrt{2} \)
- **Sum of roots**: \( 4 \)
- **Product of roots**: \( -14 \)

Do you have any questions, or would you like a detailed explanation of any specific step?

### Related Questions:
1. How do you perform synthetic division with a polynomial?
2. What is the quadratic formula, and how is it derived?
3. How can you verify if a number is a root of a given polynomial?
4. What is the significance of the sum and product of roots in a polynomial?
5. How can you use factoring techniques to solve higher-degree polynomials?

### Tip:
Always double-check your synthetic division process to ensure no errors in the factorization steps.

Learn how to solve the polynomial equation x^3 - 4x^2 + cx + 14 step-by-step with a detailed solution. This problem involves finding the coefficient c using a known root, verifying the factored form, and calculating the sum and product of roots. Suitable for advanced high school students.

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