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To solve the given polynomial division problems using the Remainder Theorem, we will substitute values for \( x \) from the binomials into the polynomial and calculate the remainder. Here's the breakdown:

### 1. \( P(x) = x^3 + 2x^2 - 4x - 8 \)

#### a. Dividing by \( x - 2 \):
To apply the Remainder Theorem, we substitute \( x = 2 \) into the polynomial.

\[
P(2) = (2)^3 + 2(2)^2 - 4(2) - 8 
\]
\[
P(2) = 8 + 2(4) - 8 - 8 
\]
\[
P(2) = 8 + 8 - 8 - 8 = 0
\]

**Remainder: 0**

Since the remainder is 0, \( x - 2 \) is a factor of \( P(x) \).

#### b. Dividing by \( x + 1 \):
Here, we substitute \( x = -1 \) into the polynomial.

\[
P(-1) = (-1)^3 + 2(-1)^2 - 4(-1) - 8 
\]
\[
P(-1) = -1 + 2(1) + 4 - 8 
\]
\[
P(-1) = -1 + 2 + 4 - 8 = -3
\]

**Remainder: -3**

Since the remainder is not 0, \( x + 1 \) is **not** a factor of \( P(x) \).

---

### 3. \( P(x) = 3x^3 - 19x^2 + 33x - 9 \)

#### a. Dividing by \( x - 3 \):
Substitute \( x = 3 \) into the polynomial.

\[
P(3) = 3(3)^3 - 19(3)^2 + 33(3) - 9 
\]
\[
P(3) = 3(27) - 19(9) + 33(3) - 9 
\]
\[
P(3) = 81 - 171 + 99 - 9 
\]
\[
P(3) = 81 - 171 + 99 - 9 = 0
\]

**Remainder: 0**

Since the remainder is 0, \( x - 3 \) is a factor of \( P(x) \).

#### b. Dividing by \( x - 1 \):
Substitute \( x = 1 \) into the polynomial.

\[
P(1) = 3(1)^3 - 19(1)^2 + 33(1) - 9 
\]
\[
P(1) = 3(1) - 19(1) + 33(1) - 9 
\]
\[
P(1) = 3 - 19 + 33 - 9 = 8
\]

**Remainder: 8**

Since the remainder is not 0, \( x - 1 \) is **not** a factor of \( P(x) \).

---

### Summary of Results:

1. \( P(x) = x^3 + 2x^2 - 4x - 8 \)
   - \( x - 2 \): **Remainder = 0** (Factor)
   - \( x + 1 \): **Remainder = -3** (Not a Factor)

3. \( P(x) = 3x^3 - 19x^2 + 33x - 9 \)
   - \( x - 3 \): **Remainder = 0** (Factor)
   - \( x - 1 \): **Remainder = 8** (Not a Factor)

---

Do you need more details or have any questions?

Here are 5 related questions:
1. How is the Remainder Theorem used to determine if a binomial is a factor of a polynomial?
2. Can synthetic division also verify whether a binomial is a factor of a polynomial?
3. What is the connection between the Remainder Theorem and the Factor Theorem?
4. How can you factor higher-degree polynomials after using the Remainder Theorem?
5. How do you handle complex roots when applying the Remainder Theorem?

**Tip:** When using the Remainder Theorem, always remember to substitute the opposite sign from the binomial divisor into the polynomial.

Find the remainder when the given polynomial is divided by each binomial and determine whether the binomial is a factor.

Learn how to find the remainder when dividing polynomials by binomials using the Remainder Theorem. This step-by-step guide includes problems involving polynomials such as x^3 + 2x^2 - 4x - 8 and 3x^3 - 19x^2 + 33x - 9.

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