## Long Division of Polynomials: (7x^3 + x^2 + x) / (x^2 + 1)

Long division of polynomials is a method used to divide two polynomials, much like long division with numbers. In this article, we'll step through the process of dividing (7x³ + x² + x) by (x² + 1).

### Step 1: Set up the Division

First, set up the long division problem. Write the dividend (7x³ + x² + x) inside the division symbol and the divisor (x² + 1) outside.

```
________
x² + 1 | 7x³ + x² + x
```

### Step 2: Divide the Leading Terms

Focus on the leading terms of the dividend and the divisor: 7x³ and x².

- Ask yourself: "What do I multiply x² by to get 7x³?"
- The answer is 7x.

Write 7x above the division symbol, aligned with the x² term.

```
7x______
x² + 1 | 7x³ + x² + x
```

### Step 3: Multiply and Subtract

Multiply the divisor (x² + 1) by the quotient term (7x):

- (x² + 1) * (7x) = 7x³ + 7x

Write this result below the dividend:

```
7x______
x² + 1 | 7x³ + x² + x
7x³ + 7x
```

Subtract this result from the dividend:

```
7x______
x² + 1 | 7x³ + x² + x
7x³ + 7x
-------
x² - 6x
```

### Step 4: Bring Down the Next Term

Bring down the next term of the dividend (which is +x):

```
7x______
x² + 1 | 7x³ + x² + x
7x³ + 7x
-------
x² - 6x + x
```

### Step 5: Repeat Steps 2-4

Now repeat steps 2-4 with the new polynomial (x² - 6x + x):

- Focus on the leading terms: x² and x².
- What do you multiply x² by to get x²? The answer is 1.
- Write 1 above the division symbol, aligned with the x term:

```
7x + 1___
x² + 1 | 7x³ + x² + x
7x³ + 7x
-------
x² - 6x + x
```

- Multiply (x² + 1) * (1) = x² + 1 and subtract it from the current polynomial:

```
7x + 1___
x² + 1 | 7x³ + x² + x
7x³ + 7x
-------
x² - 6x + x
x² + 1
-----
-6x
```

### Step 6: The Remainder

We can't divide any further since the degree of the remaining polynomial (-6x) is less than the degree of the divisor (x²).

**Therefore, the result of dividing (7x³ + x² + x) by (x² + 1) is 7x + 1 with a remainder of -6x.**

We can write this in the form:

**(7x³ + x² + x) / (x² + 1) = 7x + 1 - (6x)/(x² + 1)**