## Long Division of Polynomials: (8x^3 - 3x + 1) / (4x^3 + x^2 - 2x - 3)

Long division of polynomials is a method for dividing one polynomial by another polynomial. It is similar to the long division of numbers. In this article, we will explore how to divide the polynomial **(8x^3 - 3x + 1)** by **(4x^3 + x^2 - 2x - 3)**.

### Setting up the Division

**Arrange the polynomials in descending order of their exponents.**In our case, both polynomials are already arranged.**Write the dividend (8x^3 - 3x + 1) under the division symbol and the divisor (4x^3 + x^2 - 2x - 3) outside the symbol.**

```
_________
4x^3 + x^2 - 2x - 3 | 8x^3 - 3x + 1
```

### Performing the Division

**Divide the leading term of the dividend (8x^3) by the leading term of the divisor (4x^3).**This gives us**2**. Write this quotient above the dividend.**Multiply the quotient (2) by the entire divisor (4x^3 + x^2 - 2x - 3).**This gives us**8x^3 + 2x^2 - 4x - 6**. Write this result below the dividend.**Subtract the result from the dividend.****Bring down the next term of the dividend (1).**

```
2
4x^3 + x^2 - 2x - 3 | 8x^3 - 3x + 1
-(8x^3 + 2x^2 - 4x - 6)
--------------------
-2x^2 + x + 7
```

**Repeat steps 1-4 with the new dividend (-2x^2 + x + 7).**

- Divide the leading term of the new dividend (-2x^2) by the leading term of the divisor (4x^3). This gives us
**-1/2x**. - Multiply the new quotient (-1/2x) by the divisor. This gives us
**-2x^2 - 1/2x + x + 3/2**. - Subtract this result from the current dividend.
- Bring down the next term (which is 0, since the original dividend didn't have an x^2 term).

```
2 - 1/2x
4x^3 + x^2 - 2x - 3 | 8x^3 - 3x + 1
-(8x^3 + 2x^2 - 4x - 6)
--------------------
-2x^2 + x + 7
-(-2x^2 - 1/2x + x + 3/2)
--------------------------
3/2x + 11/2
```

**Continue this process until the degree of the remaining dividend is less than the degree of the divisor.**In this case, the degree of (3/2x + 11/2) is 1, which is less than the degree of (4x^3 + x^2 - 2x - 3). Therefore, we stop here.

### The Result

The result of the long division is:

**(8x^3 - 3x + 1) / (4x^3 + x^2 - 2x - 3) = 2 - 1/2x + (3/2x + 11/2) / (4x^3 + x^2 - 2x - 3)**

This can also be written as:

**(8x^3 - 3x + 1) = (4x^3 + x^2 - 2x - 3)(2 - 1/2x) + (3/2x + 11/2)**

This indicates that the original polynomial (8x^3 - 3x + 1) can be expressed as the product of the divisor (4x^3 + x^2 - 2x - 3) and the quotient (2 - 1/2x) plus the remainder (3/2x + 11/2).