Long Division vs Synthetic Division of Polynomials.

 Most math teachers end up teaching both long division and synthetic division of polynomials at some point in the classes.  I know that most of my students never take time to ask why they need to learn both when synthetic is easier.

They do not recognize that one is only used with an expression to the first power while the other can be used with anything.  The two are related.

Long division can be used to divide one polynomial by another polynomial that is of a lesser degree.  You can use long division to divide a problem such as x^4 + 3x^3 - 2/x^3 + 4.
You don't need to rewrite anything.

Synthetic division, on the other hand, only works when you are dividing a linear term (degree one) into a higher degree polynomial.  So this one is restricted to a very limited pool of polynomials.

This is one reason why you cannot use synthetic division for every single problem unless you can easily factor the polynomial you are dividing into the other polynomial such as x^2 - 1 or x^2 +4x + 4.  These you can quickly factor those to get the degree one polynomial so you can use synthetic division.

The other problem with using synthetic division instead of long division is that it sometimes becomes much harder when you have something like 3x + 2 = 0, you have to rewrite it to x = -2/3 so you are. using fractions where as when you do the division using long division, you do not have to rewrite anything and do not have to work with fractions.

I teach both ways but I haven't always taken time to explain why you might use long division rather than synthetic division even though it is easier for people to do.  Let me know what you think, I'd love to hear.  Have a great day.  This is a bit short but I've been working at a conference and didn't have a lot of time.  I hope to get back to normal tomorrow.