use a synthetic division table to solve $$ \frac{x^3+6x^2-2x-12}{x+6} $$

To solve $\frac{x^3 + 6x^2 - 2x - 12}{x + 6}$ using synthetic division, we will follow these steps:

1. Identify the coefficients of the polynomial $x^3 + 6x^2 - 2x - 12$.
2. Use the zero of the divisor $x + 6$ which is $-6$.
3. Perform synthetic division.

### Step-by-Step Process

1. **Identify the coefficients:**
   The polynomial is $x^3 + 6x^2 - 2x - 12$.
   - Coefficients: $1, 6, -2, -12$

2. **Setup synthetic division:**
   - Use the zero of $x + 6$, which is $-6$.

3. **Perform synthetic division:**
   
   \[
   \begin{array}{r|rrrr}
     -6 & 1 & 6 & -2 & -12 \\
       &   & -6 & 0 & 12 \\
     \hline
       & 1 & 0 & -2 & 0 \\
   \end{array}
   \]

### Explanation

1. **Write down the coefficients** of the polynomial: $1, 6, -2, -12$.
2. **Bring down the first coefficient**: $1$.
3. **Multiply** $-6$ (the zero) by the number just written below the line and **write the product** under the next coefficient.
4. **Add** the numbers in the second column: $6 + (-6) = 0$.
5. **Multiply** $-6$ by the result obtained in the previous step: $-6 \times 0 = 0$, and **write it** under the next coefficient.
6. **Add** the numbers in the third column: $-2 + 0 = -2$.
7. **Multiply** $-6$ by the result obtained in the previous step: $-6 \times -2 = 12$, and **write it** under the next coefficient.
8. **Add** the numbers in the fourth column: $-12 + 12 = 0$.

### Conclusion

The quotient is given by the coefficients $1, 0, -2$, which correspond to $x^2, x, \text{and the constant term}$ respectively. So, the quotient is:

\[ x^2 - 2 \]

The remainder is $0$.

### Final Answer

\[
\frac{x^3 + 6x^2 - 2x - 12}{x + 6} = x^2 - 2
\]

Do you have any questions or need further details?

### Relative Questions
1. How does synthetic division differ from long division in polynomials?
2. Can synthetic division be used for all polynomial divisions?
3. How can you check if your synthetic division result is correct?
4. What are the limitations of synthetic division?
5. How do you handle polynomials with missing terms in synthetic division?
6. Can synthetic division be applied to divisors of the form $ax + b$?
7. What is the significance of the remainder in polynomial division?
8. How can synthetic division be used to find polynomial roots?

### Tip
Always double-check the multiplication and addition steps in synthetic division to avoid errors in the final result.

Learn how to use synthetic division to solve the polynomial division problem $ \frac{x^3 + 6x^2 - 2x - 12}{x + 6} $ step-by-step. This problem covers polynomial division concepts, including synthetic division method. Suitable for high school students.

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