Long Polynomial Division

And you didn't guess the whole answer right away; instead, you started working on the "front" part (that is, the larger place-value part) of the number you were dividing. For instance, if you were dividing 1137 by 82, you'd look at the "8" and the "10", and guess that probably a "1" should go on top, above the "11", because 8 fits once into 11.

Long division for polynomials works in much the same way:

First, I'll set up the division, putting the dividend (the thing being divided into) inside and the divisor (the thing doing the dividing) outside and to the left:

For the moment, I'll ignore the everything past the leading terms. Just as with numerical long division, I will look just at the leading x of the divisor and the leading x² of the dividend.

If I divide the leading x² inside by the leading x in front, what would I get? I'd get an x. So I'll put an x on top of the division symbol, right above the x² inside:

Now I'll take that x on top, and I'll multiply it through the divisor, x + 1. First, I'll multiply the x (on top) by the x (on the "side"), and carry the resulting x² underneath, putting it directly below the x² from the dividend:

Then I'll multiply the x (on top) by the 1 (on the "side"), and carry the 1x underneath, putting it directly below the −9x in the dividend:

Then I'll draw the horizontal "equals" bar underneath what I've just put underneath the dividend, so I can do the subtraction.

To subtract the polynomials, I first change all the signs in the second line...

...and then I add down. The first term (the x²) will cancel out (by design), while the −9x − 1x becomes −10x:

I need to remember to carry down that last term (that is, the "subtract ten" term) from the dividend:

At this point, I start ignoring the dividend, and instead work on the bottom line of my long division.

I look at the x from the divisor and the new leading term, the –10x, in the bottom line of the division. If I divide the −10x by the x, I would end up with a −10, so I'll put that on top, right above the −9x:

Now I'll multiply the −10 (on top) by the leading x (on the "side"), and carry the −10x to the bottom, directly underneath the previous line's −10x:

...and I'll multiply the −10 (on top) by the 1 (on the "side"), and carry the −10 to the bottom, directly below the previous line's −10:

I'll draw another horizontal "equals" bar, and change the signs on all the terms in the bottom row:

Then I add down:

By design, the 10x's cancelled off. By happenstance, the 10's cancelled off, too. Then my answer, from across the top of the division symbol, is:

This fraction-reduction can be done in either of two ways: I can factor the quadratic and then cancel the common factor, like this:

But what if I didn't know how to factor (or if I have to "show my work" for the long polynomial division on a test)? As previously, I'll start the long division by working with the leading terms of the divisor and the dividend.

The leading term of the dividend is x² and the leading term of the divisor is x. Dividing x² by x gives me x, so that's what I put up on top, directly over the x² in the dividend:

Then I multiply the x on top onto the divisor x + 7, and put the resulting x² + 7 underneath the dividend:

Then I draw the horizontal "equals" bar, change the signs, add down,and carry the +14 down, getting 2x + 14 under the "equals" bar:

Dividing the leading 2x by the divisor's leading x gives me 2, so that's what I put on top of the division symbol, right above the 9x in the dividend:

Then I multiply this 2 on top against the x + 7, and put the result, 2x + 14, underneath:

Then I change the signs, and add down, getting a zero remainder:

The answer to the division is the quotient, being the polynomial across the top of the long-division symbol: