How to depress a general polynomial

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 nth 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 nth degree polynomial

Here a is not zero, or else we wouldn't have an nth degree polynomial.
The following calculation shows that the change of variables

results in an nth 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 3a may not be invertible in a field of characteristic 2 or 3.

Related posts

• How to solve a cubic equation
• How to remove the first order term from a second order ODE

The post How to depress a general polynomial first appeared on John D. Cook.