Synthetic Division Example and Steps Explained

Problem:

The task is to divide the polynomial \( x^3 - 2x^2 - 8x - 35 \) by \( x - 5 \) using synthetic division. The steps are outlined in the image, and we will follow them to solve the problem.

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Step-by-Step Solution:

#### 1. Identify the known zero:
The divisor is \( x - 5 \), so the known zero is \( 5 \). This value will be used in the synthetic division process.

#### 2. List the coefficients of the dividend:
The dividend is \( x^3 - 2x^2 - 8x - 35 \). The coefficients of the terms are:
\[ 1, -2, -8, -35 \]
These correspond to the coefficients of \( x^3 \), \( x^2 \), \( x \), and the constant term, respectively.

#### 3. Set up the synthetic division:
Write the known zero \( 5 \) to the left of a vertical line, and write the coefficients of the dividend to the right of the vertical line:
\[
\begin{array}{r|rrrr}
5 & 1 & -2 & -8 & -35 \\
& & & & \\
\hline
& & & & \\
\end{array}
\]

#### 4. Bring down the first coefficient:
Bring down the first coefficient (1) below the horizontal line:
\[
\begin{array}{r|rrrr}
5 & 1 & -2 & -8 & -35 \\
& & & & \\
\hline
& 1 & & & \\
\end{array}
\]

#### 5. Multiply the first coefficient by the known zero:
Multiply the brought-down coefficient (1) by the known zero (5):
\[ 1 \times 5 = 5 \]
Write this result (5) under the next coefficient (-2):
\[
\begin{array}{r|rrrr}
5 & 1 & -2 & -8 & -35 \\
& & 5 & & \\
\hline
& 1 & & & \\
\end{array}
\]

#### 6. Add the next coefficient and the product:
Add the next coefficient (-2) and the product (5):
\[ -2 + 5 = 3 \]
Write this result (3) below the horizontal line:
\[
\begin{array}{r|rrrr}
5 & 1 & -2 & -8 & -35 \\
& & 5 & & \\
\hline
& 1 & 3 & & \\
\end{array}
\]

#### 7. Repeat the process:
- Multiply the last result (3) by the known zero (5):
\[ 3 \times 5 = 15 \]
Write this result (15) under the next coefficient (-8):
\[
\begin{array}{r|rrrr}
5 & 1 & -2 & -8 & -35 \\
& & 5 & 15 & \\
\hline
& 1 & 3 & & \\
\end{array}
\]

- Add the next coefficient (-8) and the product (15):
\[ -8 + 15 = 7 \]
Write this result (7) below the horizontal line:
\[
\begin{array}{r|rrrr}
5 & 1 & -2 & -8 & -35 \\
& & 5 & 15 & \\
\hline
& 1 & 3 & 7 & \\
\end{array}
\]

#### 8. Repeat again:
- Multiply the last result (7) by the known zero (5):
\[ 7 \times 5 = 35 \]
Write this result (35) under the next coefficient (-35):
\[
\begin{array}{r|rrrr}
5 & 1 & -2 & -8 & -35 \\
& & 5 & 15 & 35 \\
\hline
& 1 & 3 & 7 & \\
\end{array}
\]

- Add the next coefficient (-35) and the product (35):
\[ -35 + 35 = 0 \]
Write this result (0) below the horizontal line:
\[
\begin{array}{r|rrrr}
5 & 1 & -2 & -8 & -35 \\
& & 5 & 15 & 35 \\
\hline
& 1 & 3 & 7 & 0 \\
\end{array}
\]

#### 9. Interpret the results:
- The numbers below the horizontal line, except the last one, represent the coefficients of the quotient polynomial.
- The last number (0) is the remainder.

Thus, the quotient is:
\[ x^2 + 3x + 7 \]
and the remainder is:
\[ 0 \]

#### Final Answer:
The result of dividing \( x^3 - 2x^2 - 8x - 35 \) by \( x - 5 \) is:
\[
\boxed{x^2 + 3x + 7}
\]