## Solving (x^3 - 8x^2 + 17x - 10) / (x - 5) Using Synthetic Division

Synthetic division is a shortcut method for dividing polynomials by binomials of the form (x - a). It provides a more efficient way to find the quotient and remainder compared to long division.

Let's solve the division problem **(x^3 - 8x^2 + 17x - 10) / (x - 5)** using synthetic division:

**Step 1: Set Up the Division**

Write down the coefficients of the dividend (x^3 - 8x^2 + 17x - 10) and the divisor (x - 5). Remember to include any missing terms with a coefficient of 0.

```
5 | 1 -8 17 -10
|__________________
```

**Step 2: Bring Down the Leading Coefficient**

Bring down the leading coefficient of the dividend (1) below the horizontal line.

```
5 | 1 -8 17 -10
|__________________
1
```

**Step 3: Multiply and Add**

Multiply the number you just brought down (1) by the divisor (5). Write the product (5) below the next coefficient of the dividend (-8). Add the two numbers (-8 + 5 = -3) and write the sum below the line.

```
5 | 1 -8 17 -10
|__________________
1 -3
```

**Step 4: Repeat the Process**

Repeat steps 2 and 3 for the remaining coefficients.

- Multiply the last result (-3) by the divisor (5), giving -15.
- Add -15 to the next coefficient (17), resulting in 2.

```
5 | 1 -8 17 -10
|__________________
1 -3 2
```

- Multiply 2 by the divisor (5), giving 10.
- Add 10 to the last coefficient (-10), resulting in 0.

```
5 | 1 -8 17 -10
|__________________
1 -3 2 0
```

**Step 5: Interpret the Results**

The numbers below the line represent the coefficients of the quotient polynomial, starting from the highest power of x. The last number (0) is the remainder.

Therefore, the quotient is **x^2 - 3x + 2** and the remainder is **0**.

**Final Answer:**

(x^3 - 8x^2 + 17x - 10) / (x - 5) = **x^2 - 3x + 2**

This means that **(x^3 - 8x^2 + 17x - 10) = (x - 5)(x^2 - 3x + 2)**.