Polynomial Long Division

Polynomial long division involves the same concepts and process as arithmetic long division.

I’ll try to explain the process with the following division.

*Whatever is different from the previous step has been written in red.

1. The setup for polynomial long division is the same as for arithmetic long division. You have a dividend to the right, a divisor to the left and a quotient on the top.
For the problem above, it will look like this.
Notice that the terms are organized in order of decreasing degree (x^3 goes first, then x^2, and then -42).
However, we need to include all the terms from x^0 to the biggest term in the dividend for the dividend (in this case, from x^3 to x^0) and in the divisor for the divisor (in this case, from x^1 to x^0) even if their coefficient is 0.

For instance, in the dividend you will find that there is no x^2. That is because its coefficient is 0. However, you still have to write it out, what means that the problem will actually look like this.
2. Now, we can start dividing.

We will start by dividing the biggest term in the dividend by the biggest term in the divisor. In this case, we will divide x^3 by x, which gives us x^2. This result, we will write on top of the dividend like we will do in a regular division.
You might want to place the term above its corresponding term for better organization. In this case, the x^2 in our quotient above the -12x^2 in our dividend.

3. Now, just like in regular long division, we will multiply the term we just wrote down (x^2) by the divisor (x-3).

x^2*(x-3) = x^3-3x^2

4. We will now subtract this product from the dividend like we normally do in arithmetic long division.
*Remember to properly distribute the negative sign.
5. Now, like in a regular division, we will bring the next term down.
6. We now have to repeat steps 2-5, only this time we will use the new polynomial we have on the bottom of our problem, like in a regular division, until we have no more terms to bring down. *Here we can see why we must also include the terms whose coefficients are 0. If not, we would end up subtracting -27x from -42, which is not possible.

Continuing with our problem…

7. Once we have no more terms to bring down from our dividend, we are done. Whatever we have left over on the bottom of our problem is the remainder, as in regular division.

To properly express our answer, we must write the quotient we have on the top + the remainder over the divisor. In this case, our answer will be:

(x^2-9x-27) + [(-123) ÷ (x-3)]

We are done!

If you still want more practice, you might want to do one of the following problems (Answers on the bottom):

Answers:
1) x-5; 2) x^2+4x+3; 3) -x^3+x^2+5x-5 + [7 ÷ (x-5)]