Polynomial long division

At school everybody is taught this procedure to divide one number by another. The first number being called the dividend and the second the divisor:

It's possible that some people reverse the order, doing it from right to left. I have no idea about what causes this, but I remember making this mistake more than once. Could it be related to languages that are written right to left? I really don't know. If you take another look at it, this algorithm is nothing more than: 3 x 100 = 300, then 523 - 300 = 223. Then 3 x 70 = 210 and 100 + 70 = 170. Repeat. It's all based on base 10. What we do is to break down numbers in base 10, such as 523 = 500 + 20 + 3. Perhaps some people think it reversed, 3 + 20 + 500 = 523? It's really like asking: "What is easier? To do the sum from 1 to 10 or from 10 to 1?". For computers this doesn't matter at all.

Polynomial long division

The first method is really the same method that everybody learns to process with plain numbers:

The result is: [math]\displaystyle{ (x - 3)(x^2 + x + 3) + 5 }[/math]. The most common mistakes here is to confuse powers with coefficients, such as the cubic power with the constant 3. The other is to divide wrong terms. We divide term by term, highest power by highest power in the same way that we divided 523 by 3 as the previous example. 5 by 3 yields 1, because 1 times 3 is the highest we can go without surpassing 5. Then the difference 5 minus 3 yields 2. We divide [math]\displaystyle{ \frac{x^3}{x} = x^2 }[/math], then [math]\displaystyle{ x^2(x - 3) = x^3 - 3x^2 }[/math]. The next mistake is that we divided [math]\displaystyle{ \frac{x^3}{x} }[/math], but we multiplied [math]\displaystyle{ x^2(x - 3) }[/math], not [math]\displaystyle{ (x^2)(x) }[/math] alone.

The wikipedia's article is a bit confusing because the divisor [math]\displaystyle{ x - 3 }[/math] was written to the left and the quotient is above the dividend. Maybe some people find it easier to read that way, I don't know.