How do you do long division with polynomials?

EX:

x-2 divided by x^3 + x^2 +4x ???

The method is described here:

http://www.purplemath.com/modules/synthdiv.htm

It is like long division except that you divide, multiply and "bring down" algebraic terms instead of numbers

To do long division with polynomials, follow these steps:

Step 1: Arrange the polynomials in descending order of degree. In this case, x^3 + x^2 + 4x should be on the top, and x - 2 should be on the bottom.

Step 2: Divide the first term of the numerator (x^3) by the first term of the denominator (x). This gives you x^2, which should be written above the division line.

Step 3: Multiply the divisor (x - 2) by the quotient you obtained in the previous step (x^2). This gives you x^3 - 2x^2.

Step 4: Subtract this result (x^3 - 2x^2) from the numerator (x^3 + x^2 + 4x). Write the result (3x^2 + 4x) below the line.

```
______
x - 2 | x^3 + x^2 + 4x
```
x^2

Step 5: Repeat steps 2 to 4 with the new polynomial (3x^2 + 4x) as the numerator.

Step 6: Divide the first term of the new numerator (3x^2) by the first term of the denominator (x). This gives you 3x, which should be written above the division line.

Step 7: Multiply the divisor (x - 2) by the quotient you obtained in the previous step (3x). This gives you 3x^2 - 6x.

Step 8: Subtract this result (3x^2 - 6x) from the new numerator (3x^2 + 4x). Write the result (10x) below the line.

```
_______
x - 2 | x^3 + x^2 + 4x
x^2 + 3x
```
3x

Step 9: Repeat steps 6 to 8 with the new polynomial (10x) as the numerator.

Step 10: Divide the first term of the new numerator (10x) by the first term of the denominator (x). This gives you 10, which should be written above the division line.

Step 11: Multiply the divisor (x - 2) by the quotient you obtained in the previous step (10). This gives you 10x - 20.

Step 12: Subtract this result (10x - 20) from the new numerator (10x). Write the result (20) below the line.

```
_______
x - 2 | x^3 + x^2 + 4x
x^2 + 3x + 10
10
```
10

Step 13: There are no more terms left in the numerator, so the division is complete. The quotient is x^2 + 3x + 10, and the remainder is 20.

Therefore, x - 2 divided by x^3 + x^2 + 4x is equal to x^2 + 3x + 10 with a remainder of 20.