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: -2 & 1 & 0 & 0 & 8 \\ \end{array}$$
2. Bring the first coefficient (1) down: -2 & 1 & 0 & 0 & 8 \\ & & 1 \\ \end{array}$$
3. Multiply 1 by -2 and write the result under the next coefficient: -2 & 1 & 0 & 0 & 8 \\ & & 1 & -2 \\ \end{array}$$
4. Add the column: $0 + (-2) = -2$: -2 & 1 & 0 & 0 & 8 \\ & & 1 & -2 \\ & & & -2 \\ \end{array}$$
5. Multiply $-2$ by $-2$ and write the result under the next coefficient: -2 & 1 & 0 & 0 & 8 \\ & & 1 & -2 & 4 \\ \end{array}$$
6. Add the column: $0 + 4 = 4$: -2 & 1 & 0 & 0 & 8 \\ & & 1 & -2 & 4 \\ & & & & 4 \\ \end{array}$$
7. Multiply $4$ by $-2$ and write the result under the next coefficient: -2 & 1 & 0 & 0 & 8 \\ & & 1 & -2 & 4 & -8 \\ \end{array}$$
8. Add the column: $8 + (-8) = 0$: -2 & 1 & 0 & 0 & 8 \\ & & 1 & -2 & 4 & -8 \\ & & & & & 0 \\ \end{array}$$

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!