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Let's walk through each of the tasks on this worksheet, focusing on synthetic division.

### Problem Breakdown:
#### 1. **Write the Dividend as a Polynomial (Standard Form)**
   We are given a synthetic division setup in the top left corner, where the divisor is -1, and the coefficients of the dividend are listed as `1, 2, -1, -2`. This corresponds to a polynomial in standard form:
   \[
   x^3 + 2x^2 - x - 2
   \]

#### 2. **Write the Quotient as a Polynomial (Standard Form)**
   The synthetic division provided in the second box starts with a divisor of 4 and coefficients `3, -10, 0, -5`. After synthetic division, the quotient is formed by the coefficients `3, 2, 8, 27`, so the quotient polynomial is:
   \[
   3x^2 + 2x + 8 + \frac{27}{x-4}
   \]

#### 3. **Write the Dividend as a Product of Two Factors**
   The third problem asks us to express the dividend as a product of two factors, based on the synthetic division shown. With the coefficients `1, 2, 5, 6` and a divisor of -2, the resulting factorization is:
   \[
   (x + 2)(x^3 + x^2 + 3x + 3)
   \]

#### 4 & 5. **Fill in Missing Values for Synthetic Division**
   These problems require filling in missing values to complete the synthetic division table. I can walk you through each of these steps in detail if you'd like, but here’s an outline:
   - Problem 4: Divisor is 2; coefficients are `1, 1, 10`; after completing synthetic division, we find the missing numbers are:
     \[
     \boxed{2}, \boxed{5}, \boxed{10}
     \]
   - Problem 5: Divisor is 1; coefficients are `4, -4, 1`; after completing synthetic division, the missing numbers are:
     \[
     \boxed{-2}, \boxed{3}, \boxed{1}
     \]

#### 6-9. **Perform Synthetic Division**
   For these exercises, you are asked to divide polynomials using synthetic division. Let's solve one step-by-step:

   - **Example for #6:**
     \[
     (2x^3 + 4x^2 - 3x - 6) \div (x + 3)
     \]
     Step-by-step:
     1. Write down the coefficients of the dividend: `2, 4, -3, -6`.
     2. The divisor `x + 3` gives a value of `-3` to use in synthetic division.
     3. Apply synthetic division:
        ```
        -3 |  2   4   -3   -6
             |     -6    6   -9
             -------------------
               2  -2     3    -15
        ```
     The quotient is:
     \[
     2x^2 - 2x + 3
     \]
     with a remainder of `-15`.

---

Would you like a more detailed breakdown for any of these problems, or further assistance with the remaining questions?

### Five Follow-up Questions:
1. How do you find the quotient when dividing polynomials using synthetic division?
2. What is the difference between dividing by \(x + 3\) and \(x - 3\) in synthetic division?
3. How do you determine the remainder when performing synthetic division?
4. Can you always use synthetic division to divide polynomials?
5. What happens when the remainder is zero in synthetic division?

### Tip:
When using synthetic division, always ensure the polynomial is in standard form, and include all terms, even those with zero coefficients.

The user uploaded an image of a synthetic division worksheet, which includes exercises on writing polynomials in standard form, performing synthetic division, and completing missing values in division tables.

This worksheet provides practice for students on synthetic division, including exercises on dividing polynomials, finding missing values, and expressing polynomials as products of two factors. Suitable for Grades 9-12.

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