## Dividing Polynomials: (x^3 - 4x^2 - 17x + 6) / (x - 3)

This article will guide you through the process of dividing the polynomial **(x^3 - 4x^2 - 17x + 6)** by the binomial **(x - 3)**. We will use the **long division method** to achieve this.

### Long Division Method

**Step 1: Set up the division problem.**

Write the problem as a long division, with the dividend (x^3 - 4x^2 - 17x + 6) under the division symbol and the divisor (x - 3) outside the symbol.

```
________
x - 3 | x^3 - 4x^2 - 17x + 6
```

**Step 2: Divide the leading terms.**

Divide the leading term of the dividend (x^3) by the leading term of the divisor (x). This gives us x^2. Write x^2 above the x^2 term in the dividend.

```
x^2
x - 3 | x^3 - 4x^2 - 17x + 6
```

**Step 3: Multiply the quotient by the divisor.**

Multiply the quotient (x^2) by the divisor (x - 3). This gives us x^3 - 3x^2. Write this result below the dividend, aligning like terms.

```
x^2
x - 3 | x^3 - 4x^2 - 17x + 6
x^3 - 3x^2
```

**Step 4: Subtract.**

Subtract the result from the previous step from the dividend. This will leave us with -x^2 - 17x.

```
x^2
x - 3 | x^3 - 4x^2 - 17x + 6
x^3 - 3x^2
---------
-x^2 - 17x
```

**Step 5: Bring down the next term.**

Bring down the next term from the dividend (-17x) next to the result of the subtraction.

```
x^2
x - 3 | x^3 - 4x^2 - 17x + 6
x^3 - 3x^2
---------
-x^2 - 17x + 6
```

**Step 6: Repeat steps 2-5.**

Repeat the steps above, dividing the new leading term (-x^2) by the leading term of the divisor (x). This gives us -x. Write -x above the -17x term in the dividend.

```
x^2 - x
x - 3 | x^3 - 4x^2 - 17x + 6
x^3 - 3x^2
---------
-x^2 - 17x + 6
-x^2 + 3x
```

Subtract, bring down the next term, and repeat the process until there are no more terms to bring down.

```
x^2 - x - 14
x - 3 | x^3 - 4x^2 - 17x + 6
x^3 - 3x^2
---------
-x^2 - 17x + 6
-x^2 + 3x
---------
-20x + 6
-20x + 60
---------
-54
```

**Step 7: Interpret the result.**

The final result is the quotient **(x^2 - x - 14)** and a remainder of **-54**.

Therefore, we can express the division as:

**(x^3 - 4x^2 - 17x + 6) / (x - 3) = x^2 - x - 14 - 54/(x - 3)**