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We are given the polynomial \( p(x) = x^3 - 2x^2 - 7x - 4 \). Let's factor this polynomial and address the questions step by step.

### Factorization of the polynomial

To factor the cubic polynomial \( p(x) = x^3 - 2x^2 - 7x - 4 \), we can apply the **Rational Root Theorem** or synthetic division to find the roots. Let's find a potential rational root by testing factors of the constant term \(-4\) and the leading coefficient \(1\).

1. Test \( x = 1 \):
   \[
   p(1) = 1^3 - 2(1^2) - 7(1) - 4 = 1 - 2 - 7 - 4 = -12
   \]
   So, \( x = 1 \) is not a root.

2. Test \( x = -1 \):
   \[
   p(-1) = (-1)^3 - 2(-1)^2 - 7(-1) - 4 = -1 - 2 + 7 - 4 = 0
   \]
   So, \( x = -1 \) is a root.

Now that we know \( x + 1 \) is a factor, we can perform synthetic division to divide \( p(x) \) by \( x + 1 \).

After performing synthetic division, we get:
\[
p(x) = (x + 1)(x^2 - 3x - 4)
\]
Now, we can factor the quadratic part:
\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\]
So the complete factorization of \( p(x) \) is:
\[
p(x) = (x + 1)^2(x - 4)
\]

### Answering the questions:

- **a) Number of prime factors**: The prime factors are \( (x + 1) \) and \( (x - 4) \), so there are **2 distinct prime factors**.
  
- **b) Factor with the highest repetition**: The factor \( (x + 1) \) appears twice, so it has the highest repetition.
  
- **c) Quadratic prime factor**: There are no irreducible quadratic factors, so the correct answer here would be **none**.
  
- **d) Sum of prime factors**: The prime factors are \( x + 1 \) and \( x - 4 \). Their sum is:
  \[
  (x + 1) + (x - 4) = 2x - 3
  \]

### Do you want further details or have any questions?

Here are 5 questions based on the above explanation:
1. How can the Rational Root Theorem help in factorizing cubic polynomials?
2. What other methods can be used to factor polynomials when synthetic division is not easy?
3. Why is synthetic division useful for finding polynomial roots?
4. How can you check if a quadratic factor is irreducible?
5. Can a polynomial have both real and complex roots when factored?

**Tip**: When factoring polynomials, always look for common factors first before applying complex methods.

Dado el polinomio

Responda
a) Al factorizar p(x) se obtienen factores primos.

b) El factor primo de mayor repitencia es.

c) El factor primo cuadrático es.

d) La suma de factores primos es.

Learn how to factorize the polynomial p(x) step by step and answer questions about prime factors, their repetitions, quadratic factors, and the sum of the factors. This problem covers key algebraic concepts such as factorization and the Rational Root Theorem, suitable for students in Grades 9-12.

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