How to do long division of polynomials?

• Divide 3x³ – 5x² + 10x – 3  by  3x + 1

This division did not come out even. What am I supposed to do with the remainder? Think back to when you did long division with plain numbers. Sometimes there would be a remainder; for instance, if you divide 132 by 5:

...there is a remainder of 2. Remember how you handled that? You made a fraction, putting the remainder on top of the divisor, and wrote the answer as "twenty-six and two-fifths":

The first form, without the "plus" in the middle, is how "mixed numbers" are written, but the meaning of the mixed number is actually the addition. We do the same thing with polynomial division. Since the remainder is –7 and since the divisor is 3x + 1, then I'll turn the remainder into a fraction (the remainder divided by the original divisor), and add this fraction to the polynomial across the top of the division symbol. Since the division looks like this:

...then the answer is this:   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
 

Warning: Do not write the polynomial "mixed number" in the same format as numerical mixed numbers! If you just append the fractional part to the polynomial part, this will be interpreted as polynomial multiplication, which is not what you mean!

Note: Different books format the long division differently. When writing the expressions across the top of the division, some books will put the terms above the same-degree term, rather than above the term being worked on. In such a text, the long division above would be presented as shown here: The only difference is that the terms across the top are shifted to the right. Otherwise, everything is exactly the same. You should probably use the formatting that your instructor uses.

• Divide 2x³ – 9x² + 15  by  2x – 5

First off, I note that there is a gap in the degrees of the terms of the dividend: the polynomial 2x³ – 9x² + 15 has no x term. My work could get very messy inside the division symbol, so it is important that I leave space for a x-term column, just in case. I can create this space by turning the dividend into 2x³ – 9x² + 0x + 15. This is a legitimate mathematical step: since I've only added zero, I haven't actually changed the value of anything. Now that I have all the "room" I might need for my work, I'll do the division:

I need to remember to add the remainder to the polynomial part of the answer:

• Divide 4x⁴ + 3x³ + 2x + 1  by x² + x + 2

I'll add a 0x² term to the dividend (inside the division symbol) to make space for my work, and then I'll do the division in the usual manner:

Then my answer is:
 

To succeed with polyomial long division, you need to write neatly, remember to change your signs when you're subtracting, and work carefully, keeping your columns lined up properly. If you do this, then these exercises should not be very hard; annoying, maybe, but not hard.