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Let's solve the division of the polynomial \( x^4 - 4x^3 + 12x^2 - 13x - 5 \) by \( x - 2 \) using synthetic division.

### Step-by-step process:

1. **Set up the synthetic division:**
   - We are dividing by \( x - 2 \), so we use \( 2 \) as the divisor.
   - The coefficients of the polynomial \( x^4 - 4x^3 + 12x^2 - 13x - 5 \) are: \( [1, -4, 12, -13, -5] \).

2. **Perform synthetic division:**
   - Write the divisor \( 2 \) on the left, and the coefficients \( [1, -4, 12, -13, -5] \) on the right.
   
   The synthetic division table will look like this:
   
   ```
   2 |  1   -4    12   -13   -5
        |      2   -4    16     6
        ----------------------------
          1   -2     8     3     1
   ```

3. **Interpret the result:**
   - The quotient is \( x^3 - 2x^2 + 8x + 3 \).
   - The remainder is \( 1 \).

### Final Answer:
- **Quotient**: \( x^3 - 2x^2 + 8x + 3 \)
- **Remainder**: \( 1 \)

Would you like more details on this process or have any questions?

Here are 5 related questions you might find helpful:
1. How is synthetic division different from long division of polynomials?
2. What are the conditions under which synthetic division can be used?
3. Can synthetic division be used for dividing by polynomials of degree higher than 1?
4. How can the remainder theorem be applied to check the remainder from synthetic division?
5. What happens when the remainder of a polynomial division is zero?

**Tip**: Always check the divisor carefully in synthetic division, as a slight error there can lead to incorrect results.

Use synthetic division to divide \( x^4 - 4x^3 + 12x^2 - 13x - 5 \) by \( x - 2 \).

Learn how to use synthetic division to divide the polynomial x^4 - 4x^3 + 12x^2 - 13x - 5 by x - 2. This solution provides a step-by-step guide to calculating the quotient and remainder, covering core concepts in polynomial division. Suitable for high school algebra students.

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