I'm trying to understand why polynomial long division works and I've hit a wall when trying to understand why we multiply all terms of the divisor by the partial quotient. Consider:
$$\frac{x^2 + 3x + 2}{x + 2}$$
During the first step we divide ${x^2}$ by ${x}$ giving us a partial quotient of ${x}$. Next, we multiply the partial quotient by the first term of the divisor, giving us ${x^2}$. So far so good, but this next step is what I don't understand: why do we then multiply the next term of the divisor by the partial quotient (i.e. ${x}$ * ${2}$)? It seems as though we're testing to see if ${x}$ can be divided into the first term of the dividend, and if it can, then we distribute the result over the whole divisor which is then subtracted. I don't understand how we can do that when we're only testing the divisibility of those first terms.
I tried a slightly different example to see what happens which highlights what I mean: $$\frac{x^2}{x + 1}$$
First, we test the divisibilty of ${x^2}$ by ${x}$. Obviously it goes ${x}$ times, but then I wanted to see if I could carry on and multiply ${x}$ * ${1}$. I carried on and the quotient becomes ${x + \frac{1}{x + 1} – 1}$. Checking $({x + 1})({x + \frac{1}{x + 1} – 1})$ does give the original dividend of ${x^2}$. Having worked through that problem I just can't see why we're able to do the multiplication of the second term of the divisor, subtract it and get everything to hold true. Essentially, if I subtract ${x^2}$ from ${x^2}$ I end up with nothing. So the second multiplication, to me at least, seems unclear as to why it works and what the purpose is.
I know I'm missing something simple here but I can't seem to make the connection. Could someone explain this to me please?
Best Answer
Think about polynomial long division in much the same way that you think about division of real numbers.
For instance, say you want to divide 48820 by 28. In a sense, doing division in this way is "greedy" -- you divide by the largest possible parts first, then work down from there.
Now, , when you long divide by 28, you first see if 28 goes into the leading term, 4. It does not, so then you check if it goes into 48 (which it does exactly once). Then, when you "bring down" the next term, you multiply the "partial quotient" -- 1 -- by 28. Right?
Now think of 28 as 2*10+8. Same operations hold.
Now, replace 10 with x. In order to keep the algorithm the same, you have to multiply by the whole denominator.
Edit: To expand a little...
Consider the previous example. 48820/28 = 1743+16/28.
Now, let's compute $\frac{4x^4+8x^3+8x^2+2x+0}{2x+8}$.
By doing polynomial long division, we obtain: $2x^3-4x^2+20x-79+\frac{632}{2x+8}$. Now you might say, "wait a minute, that has a leading coefficient of 2, but obviously 28 goes into 48 only once!" Yes, this is true, but notice that we have minus signs here. That is key.
Now, let's set $x = 10$ and see what we compute: $2000-400+200-79+\frac{632}{28}$. Doing the arithmetic, we see this is exactly our desired result.