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To solve the problem of dividing \( x^3 + 8 \) by \( x + 2 \) using synthetic division, let's go step-by-step.

### Step 1: Set up for synthetic division.
Synthetic division works when the divisor is in the form \( x - c \). Here, we have \( x + 2 \), which can be written as \( x - (-2) \). So, \( c = -2 \).

We will divide \( x^3 + 0x^2 + 0x + 8 \) (since there are missing terms for \( x^2 \) and \( x \)) by \( x + 2 \).

The coefficients of \( x^3 + 0x^2 + 0x + 8 \) are:  
\[ [1, 0, 0, 8] \]

### Step 2: Perform synthetic division.

1. Write \( c = -2 \) on the left and the coefficients on the right:
   \[
   \begin{array}{r|rrrr}
   -2 & 1 & 0 & 0 & 8 \\
   \end{array}
   \]
   
2. Bring the first coefficient (1) down:
   \[
   \begin{array}{r|rrrr}
   -2 & 1 & 0 & 0 & 8 \\
     &   & 1 \\
   \end{array}
   \]
   
3. Multiply 1 by -2 and write the result under the next coefficient:
   \[
   \begin{array}{r|rrrr}
   -2 & 1 & 0 & 0 & 8 \\
     &   & 1 & -2 \\
   \end{array}
   \]
   
4. Add the column: \( 0 + (-2) = -2 \):
   \[
   \begin{array}{r|rrrr}
   -2 & 1 & 0 & 0 & 8 \\
     &   & 1 & -2 \\
     &   &   & -2 \\
   \end{array}
   \]
   
5. Multiply \( -2 \) by \( -2 \) and write the result under the next coefficient:
   \[
   \begin{array}{r|rrrr}
   -2 & 1 & 0 & 0 & 8 \\
     &   & 1 & -2 & 4 \\
   \end{array}
   \]
   
6. Add the column: \( 0 + 4 = 4 \):
   \[
   \begin{array}{r|rrrr}
   -2 & 1 & 0 & 0 & 8 \\
     &   & 1 & -2 & 4 \\
     &   &   &   & 4 \\
   \end{array}
   \]
   
7. Multiply \( 4 \) by \( -2 \) and write the result under the next coefficient:
   \[
   \begin{array}{r|rrrr}
   -2 & 1 & 0 & 0 & 8 \\
     &   & 1 & -2 & 4 & -8 \\
   \end{array}
   \]
   
8. Add the column: \( 8 + (-8) = 0 \):
   \[
   \begin{array}{r|rrrr}
   -2 & 1 & 0 & 0 & 8 \\
     &   & 1 & -2 & 4 & -8 \\
     &   &   &   &   & 0 \\
   \end{array}
   \]

### Step 3: Interpret the result.
The bottom row gives the coefficients of the quotient and the remainder:
- The quotient is \( x^2 - 2x + 4 \).
- The remainder is 0, so there is no remainder.

### Final Answer:
The result of dividing \( x^3 + 8 \) by \( x + 2 \) is:
\[
q(x) = x^2 - 2x + 4
\]

In the required form \( q(x) + \frac{r(x)}{d(x)} \), the answer is simply:
\[
x^2 - 2x + 4
\]

Would you like any further details on synthetic division or have any questions?

Here are some related questions you may want to explore:
1. How does synthetic division work with a non-zero remainder?
2. How would the process differ if dividing by \( x - 2 \)?
3. Can synthetic division handle polynomials of higher degrees?
4. What is the difference between synthetic and long division?
5. How do you interpret results when dividing by a divisor with a leading coefficient other than 1?

**Tip:** Synthetic division works best when the divisor is a linear binomial of the form \( x - c \), and it's a quick alternative to long division for these cases!

Use synthetic division to divide x^3 + 8 by x + 2. Write your answer in the form q(x) + r(x)/d(x), where q(x) is the quotient, r(x) is the remainder, and d(x) is the divisor.

Learn how to use synthetic division to divide the polynomial x^3 + 8 by x + 2. Follow the step-by-step process and understand the concepts of quotient, remainder, and divisor. Suitable for students in Grades 10-12.

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