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To solve the problems using the **Remainder Theorem**, we need to evaluate each polynomial at the value that makes the divisor \( Q(x) \) equal to zero.

### Given:
- \( Q(x) = x + 6 \)
- The Remainder Theorem tells us that the remainder when dividing \( P(x) \) by \( Q(x) \) can be found by evaluating \( P(x) \) at \( x = -6 \) (since \( Q(x) = x + 6 \) gives \( x = -6 \)).

Let's go through each problem:

### 1. \( P(x) = x^4 + 5x^3 + 11x^2 - 25x + 29 \)

We will substitute \( x = -6 \) into the polynomial:

\[
P(-6) = (-6)^4 + 5(-6)^3 + 11(-6)^2 - 25(-6) + 29
\]

Calculating step by step:
\[
P(-6) = 1296 + 5(-216) + 11(36) - 25(-6) + 29
\]
\[
P(-6) = 1296 - 1080 + 396 + 150 + 29
\]
\[
P(-6) = 1296 - 1080 + 396 + 150 + 29 = 791
\]

So, the remainder is **791**.

---

### 2. \( P(x) = x^4 + 11x^3 + 33x^2 + 24x + 32 \)

Again, substitute \( x = -6 \) into the polynomial:

\[
P(-6) = (-6)^4 + 11(-6)^3 + 33(-6)^2 + 24(-6) + 32
\]

Calculating step by step:
\[
P(-6) = 1296 + 11(-216) + 33(36) + 24(-6) + 32
\]
\[
P(-6) = 1296 - 2376 + 1188 - 144 + 32
\]
\[
P(-6) = 1296 - 2376 + 1188 - 144 + 32 = -4
\]

So, the remainder is **-4**.

---

Would you like further details or clarification on these steps? 

Here are 5 related questions you might find helpful:
1. What is the general formula for the Remainder Theorem?
2. How can the Remainder Theorem be used to factor polynomials?
3. What happens if the remainder is zero when applying the Remainder Theorem?
4. Can the Remainder Theorem be applied to polynomials of higher degrees?
5. How does the Remainder Theorem relate to synthetic division?

**Tip:** The Remainder Theorem is particularly useful in checking whether a number is a root of a polynomial, as the remainder will be zero if it is!

Using the Remainder Theorem, find the remainder when the given polynomial P(x) is divided by Q(x).

Learn how to apply the Remainder Theorem to find the remainder when dividing the polynomial P(x) by Q(x). This example walks through two polynomials, showcasing key algebraic steps to solve for remainders, suitable for Grades 9-11.

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