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We are given that \((x + 2)\) is a factor of the polynomial \(x^4 - 4x^3 - 7x^2 + px + 48\), and we are asked to find the value of \(p\).

Since \((x + 2)\) is a factor, by the Factor Theorem, the polynomial should equal zero when \(x = -2\). Therefore, we substitute \(x = -2\) into the polynomial and set it equal to zero:

\[
(-2)^4 - 4(-2)^3 - 7(-2)^2 + p(-2) + 48 = 0
\]

Now, let's compute each term:

\[
(-2)^4 = 16
\]
\[
-4(-2)^3 = -4(-8) = 32
\]
\[
-7(-2)^2 = -7(4) = -28
\]
\[
p(-2) = -2p
\]

Now substitute these values into the equation:

\[
16 + 32 - 28 - 2p + 48 = 0
\]

Simplify the constant terms:

\[
16 + 32 - 28 + 48 = 68
\]

So the equation becomes:

\[
68 - 2p = 0
\]

Now, solve for \(p\):

\[
2p = 68 \quad \Rightarrow \quad p = 34
\]

Thus, the value of \(p\) is \(34\).

---

Would you like further details on any step, or have any other questions? Here are some related questions to expand on this concept:

1. What is the Factor Theorem and how is it used to solve polynomial equations?
2. How do you perform polynomial division when factoring higher-degree polynomials?
3. How can the Remainder Theorem be applied to polynomials?
4. What are synthetic division and its benefits over long division in polynomial problems?
5. How would the approach change if \((x - 2)\) were a factor instead?

**Tip**: Whenever you're working with polynomials and factors, using the Factor Theorem is a quick and efficient method to find unknown constants like \(p\).

(x + 2) is a factor of x^4 - 4x^3 - 7x^2 + px + 48, where p is a constant. Find the value of p.

Learn how to determine the value of p when (x + 2) is a factor of the polynomial x^4 - 4x^3 - 7x^2 + px + 48. This problem utilizes key algebraic concepts such as the Factor Theorem. Ideal for students in Grades 9-11.

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