## Dividing Polynomials: (x^3 - 3x^2 - 5x - 25) / (x - 5)

This article explores the process of dividing the polynomial **(x^3 - 3x^2 - 5x - 25)** by the binomial **(x - 5)**. We'll use **polynomial long division** to achieve this.

### Understanding Polynomial Long Division

Polynomial long division is similar to the long division we learned in arithmetic. It involves the following steps:

**Set up:**Write the dividend (the polynomial being divided) inside the division symbol and the divisor (the polynomial dividing) outside.**Divide:**Divide the leading term of the dividend by the leading term of the divisor. Write the quotient above the dividend.**Multiply:**Multiply the quotient by the entire divisor. Write the product below the dividend.**Subtract:**Subtract the product from the dividend.**Bring down:**Bring down the next term of the dividend.**Repeat:**Repeat steps 2-5 until there are no more terms to bring down.

### Applying the Steps

Let's apply these steps to our problem:

```
x^2 + 2x + 5
____________________
x - 5 | x^3 - 3x^2 - 5x - 25
-(x^3 - 5x^2)
____________________
2x^2 - 5x
-(2x^2 - 10x)
____________________
5x - 25
-(5x - 25)
____________________
0
```

### Interpreting the Results

The result of our division is **x^2 + 2x + 5**. This means that:

**(x^3 - 3x^2 - 5x - 25) / (x - 5) = x^2 + 2x + 5**

We can also express this as:

**(x^3 - 3x^2 - 5x - 25) = (x - 5)(x^2 + 2x + 5)**

This tells us that **(x - 5)** is a factor of the polynomial **(x^3 - 3x^2 - 5x - 25)**.

### Conclusion

By using polynomial long division, we successfully divided the polynomial **(x^3 - 3x^2 - 5x - 25)** by the binomial **(x - 5)** and determined the quotient to be **x^2 + 2x + 5**. This process allows us to factorize polynomials and gain a deeper understanding of their relationships.