Long division and the "add-substract" trick

Dividing polynomials with remainder is one of the basic algebraic skills and here in calculus we use it mainly when decomposing to partial fractions. We will explain this algorithm on a suitable example.

Problem: We divide with remainder

In our example this algorithm leads to

and therefore

The "add-substract" trick

It is a very general trick that is ocassionally used when manipulating algebraic expression. Often we find ourselves in position when we would really appreciate if there was a certain something in the given expression, but we do not have it there. In many cases we can simply put it there by adding, but to balance it out we have to subtract it as well.

We will show how it can apply to the example above, this time we try to divide in another way. For instance, we would really like if we had 2x3 − 2x in the numerator, so we create it there and then use it.

Now we would really like it if we had -x2 + 1 in the numerator, so we again create it there.

This way of dividing can be sometimes really fast and convenient.

There is a similar trick that we can call "multiply-divide". For instance, imagine that we have the expression 3x + 5, but we would prefer to have the coefficent 7 at x and do not want other terms with x there. This last requirement prevents us from adding and subtracting, but we can do this:

3x + 5 = 3⋅(x + 5/3) = (3/7)⋅(7x + 35/3).