Dividing Polynomials worksheet featuring 8 problems for synthetic division practice.

The image you uploaded contains a worksheet titled "Dividing Polynomials" with problems that require solving using synthetic division. Below, I will explain the general process of synthetic division and solve one of the problems as an example.

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General Steps for Synthetic Division:

1. Set Up the Problem:
- Write the coefficients of the dividend (the polynomial being divided) in descending order of powers. Include zeros for any missing terms.
- Identify the root of the divisor. If the divisor is \( x - c \), then the root is \( c \).

2. Perform Synthetic Division:
- Write the root to the left of a vertical bar.
- Bring down the leading coefficient of the dividend.
- Multiply the root by the number just brought down, and write the result under the next coefficient.
- Add the column of numbers and write the sum below the line.
- Repeat the multiplication and addition steps until all columns are processed.

3. Interpret the Results:
- The numbers below the line represent the coefficients of the quotient, starting with the degree one less than the dividend.
- The last number is the remainder.

4. Write the Answer:
- The quotient is written as a polynomial, and the remainder (if any) is expressed as a fraction over the divisor.

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Example Problem: Solve Problem #1

#### Problem:
Divide \( 2x^3 + 5x^2 - 7x + 3 \) by \( x + 2 \) using synthetic division.

#### Step 1: Set Up the Problem
- Dividend: \( 2x^3 + 5x^2 - 7x + 3 \)
- Coefficients: \( 2, 5, -7, 3 \)
- Divisor: \( x + 2 \)
- Root: \( -2 \) (since \( x + 2 = 0 \) implies \( x = -2 \))

#### Step 2: Perform Synthetic Division

| | 2 | 5 | -7 | 3 |
|-------|------|------|------|------|
| -2 | | -4 | 18 | -22 |
|-------|------|------|------|------|
| | 2 | 1 | 11 | -19 |

- Step-by-Step Explanation:
1. Bring down the leading coefficient \( 2 \).
2. Multiply \( 2 \times (-2) = -4 \) and add to the next coefficient \( 5 \): \( 5 + (-4) = 1 \).
3. Multiply \( 1 \times (-2) = -2 \) and add to the next coefficient \( -7 \): \( -7 + (-2) = -9 \).
4. Multiply \( -9 \times (-2) = 18 \) and add to the next coefficient \( 3 \): \( 3 + 18 = 21 \).

#### Step 3: Interpret the Results
- The numbers below the line are \( 2, 1, 11, -19 \).
- The quotient is \( 2x^2 + x + 11 \).
- The remainder is \( -19 \).

#### Step 4: Write the Answer
The result of the division is:
\[
\boxed{2x^2 + x + 11 - \frac{19}{x + 2}}
\]

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Additional Notes:

If you need solutions for other problems or further clarification, feel free to ask! The same process applies to all problems involving synthetic division.