The expression x3 + ax−2x−4 divisible exactly by (x + 1). Find the value of a and the remainder when the expression is divided by (x+2)

Given the polynomial expression \( x^3 + ax - 2x - 4 \), we know that it is divisible exactly by \( (x + 1) \). This means that when the polynomial is divided by \( (x + 1) \), the remainder should be zero. We can use this information to find the value of \( a \).

### Step 1: Use the Factor Theorem

The Factor Theorem states that if \( (x + 1) \) is a factor of the polynomial \( P(x) \), then:
\[
P(-1) = 0
\]

Let's substitute \( x = -1 \) into the polynomial \( P(x) = x^3 + ax - 2x - 4 \):

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

Since \( P(-1) = 0 \) (because \( x + 1 \) is a factor), we have:

\[
-3 - a = 0
\]

So,
\[
a = -3
\]

### Step 2: Find the remainder when the polynomial is divided by \( (x + 2) \)

We now know that the polynomial is \( x^3 - 3x - 2x - 4 \), which simplifies to \( x^3 - 5x - 4 \).

The remainder when dividing \( P(x) = x^3 - 5x - 4 \) by \( (x + 2) \) can be found by substituting \( x = -2 \) into the polynomial.

\[
P(-2) = (-2)^3 - 5(-2) - 4 = -8 + 10 - 4 = -2
\]

### Final Answers:
- The value of \( a \) is **-3**.
- The remainder when the polynomial is divided by \( (x + 2) \) is **-2**.

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Here are 8 related questions you might find interesting:
1. What is the Factor Theorem, and how is it used to find roots of polynomials?
2. How do you find the remainder of a polynomial division using synthetic division?
3. Can a polynomial have more than one remainder when divided by different linear factors?
4. What is the significance of the remainder when dividing polynomials?
5. How does the remainder change if the divisor is changed to \( x - 2 \) instead of \( x + 2 \)?
6. How can you verify that a given polynomial is divisible by a linear factor?
7. What are the possible values of \( a \) if the polynomial \( x^3 + ax - 2x - 4 \) is divisible by \( x + 2 \)?
8. What happens if the divisor is a quadratic expression instead of a linear one?

**Tip**: The Remainder Theorem is a quick way to find the remainder when a polynomial is divided by a linear factor \( x - c \). Just substitute \( c \) into the polynomial and evaluate!

Explore how to solve polynomial division problems using the Factor Theorem and Remainder Theorem. This problem involves finding the value of a in a polynomial divisible by (x + 1) and calculating the remainder when divided by (x + 2). Suitable for advanced high school students.

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