This post showed how to do a change of variables to remove the quadratic term from a cubic equation. Here we will show that the technique works more generally to remove the *x*^{n-1} term from an *n*th degree polynomial.

We will use big-O notation *O*(*x*^{k}) to mean terms involving *x* to powers no higher than *k*. This is slightly unusual, because typically big-O notation is used when some variable is tending to a limit, and we’re not taking limits here.

Let’s start with an *n*th degree polynomial

Here *a* is not zero, or else we wouldn’t have an *n*th degree polynomial.

The following calculation shows that the change of variables

results in an *n*th degree polynomial in *t* with no term involving *x*^{n – 1}.

## Finite fields

This approach works over real or complex numbers. It even works over finite fields too, if you can divide by *na*.

I’ve mentioned a couple times that the Weierstrass form of an elliptic curve

is the most general except when working over a field of characteristic 2 or 3. The technique above breaks down because 3*a* may not be invertible in a field of characteristic 2 or 3.

What properties does the depressed polynomial share with the original? Are they exactly identical? Surely they must differ in some ways, and are they ever interesting ways?