Polynomial Long Division
Polynomial Long Division is a reimagining of standard long division but oriented towards polynomials. This can make it seem far more intimidating but it's the exact same process.
This post will be dedicated to outlining the entire process in a stepbystep form by solving \[(360x^2 + 221x + 3) \div (5x + 3)\]

The very first thing we need to do is identify our divisor and dividend. The divisor is the polynomial that we are using to divide while the dividend is the polynomial being divided. For \[(360x^2 + 221x + 3) \div (5x+ 3),\] you would say that the dividend is \[360x^2 + 221x + 3\] since that is the polynomial being divided, and the divisor is \[5x + 3\] since that is the polynomial being used to divide.

Now, we need to setup our long division with the dividend to the right and under the long division bracket while the divisor will be to the left and outside of it.
\[\begin{array}{r} \text{divisor} \rarr 5x + 3 {\overline{\smash{\big)}\,360x^2 + 221x + 3} \larr \text{dividend}} \end{array} \]
If you started with the long division already set up, you can undo step 2 to get the dividend and divisor

After we have set up our long division, we need to figure out how many of the divisor we need to get rid of the highest order variable (the piece of the polynomial with the largest exponent) in the dividend and put that in the long division answer section above the element we are trying to remove to build our answer up. We can do this by taking the highest order variable from the dividend and dividing it by the highest order variable from the divisor. For example, we would take \[360x^2\] from the dividend and \[5x\] from the divisor. Then, we would divide \(360x^2\) by \(5x\) to get \[\frac{360x^2}{5x}\] The factor \(360\) and the factor \(5\) would divide to become \(72\) while the factor \(x^2\) and the factor \(x\) would cancel to become \(x\) giving us our answer to step 3: \[72x\] We would then put that answer into our long division answer section above the element we are trying to get rid of
\[\begin{array}{r} 72x\phantom{+ 221x + 3} \\ 5x + 3 {\overline{\smash{\big)}\,360x^2 + 221x + 3}}\end{array}\]

Now, we need to multiply the divisor by the factor we found in step 3. In our example, that means that we are going to multiply \(5x + 3\) by \(72x\) like so \[(5x + 3) \cdot 72x = 360x^2 + 216\]

Then, we take the result from 4 and subtract it from our dividend to get rid of the highest order variable if possible. Thus, we are simplifying our dividend to make it easier to work with. In our example from above that would look like
\[\begin{array}{r} 72x\phantom{+ 221x + 3} \\ 5x + 3 {\overline{\smash{\big)}\,360x^2 + 221x + 3}} \\ \underline{360x^2  216x}\phantom{ + 3} \\ 5x + 3\end{array} \]

Finally, we check if there is any piece of the equation remaining that we need to divide (that is, is the new dividend smaller than our divisor). If there isn't any remaining pieces to divide, then we give back whatever is left as the remainder. Even if that is \(0\) If there is, we go back to step 3 but now we use the remaining piece of the equation as our new dividend. So for our example, we can see that there is \(5x + 3\) remaining and so go from step 3 to step 5 again but with \(5x + 3\) as our divisor. That would look like

Go back to step 3 but with our new dividend \(5x + 3\). The highest order element from our divisor is still \(5x\), while the highest order element from our new dividend is also \(5x\). We divide \(5x\) by \(5x\) and get \(1\) so we add that to our long division answer section above the element we are trying to get rid of.
\[\begin{array}{r} 72x + 1\phantom{21x + 3} \\ 5x + 3 {\overline{\smash{\big)}\,360x^2 + 221x + 3}} \\ \underline{360x^2  216x}\phantom{ + 3} \\ 5x + 3\end{array} \]

Go back to step 4 with our new dividend. Multiplying the divisor by 1 gives us the divisor so our result for step 4 is \[5x + 3\]

Go back to step 5 with the result we just got and subtract it from the new dividend like so
\[\begin{array}{r} 72x + 1\phantom{21x + 3} \\ 5x + 3 {\overline{\smash{\big)}\,360x^2 + 221x + 3}} \\ \underline{360x^2  216x}\phantom{ + 3} \\ 5x + 3 \\ \underline{5x  3} \\ 0 \end{array} \]

Go back to step 6 with the new dividend we just got. The new dividend is \(0\) so we know that we can stop here. Otherwise, we would have to keep going.

Ok, well that's the first post! If you have any questions, feel free to ask them in the comments or shoot me an email at john@johnmarsden.dev