Finding Degree from a Table

A polynomial's degree controls how fast it curves. A line (degree 1) changes by the same amount every step, so its first differences are constant. A parabola (degree 2) doesn't change by a constant amount, but its rate of change does — so you need to difference twice before things flatten out. This pattern keeps going: a degree n polynomial needs exactly n rounds of differencing before the row goes constant. That constant value even tells you the leading coefficient, since it equals n! times the coefficient of x^n.

To use this, take a column of outputs from equally spaced x-values. Subtract consecutive outputs to get first differences. If that row is constant, you have a line. If not, subtract consecutive entries in that row to get second differences, and keep going. The row that finally becomes constant tells you the degree — if it takes three rounds, the polynomial is degree 3.

This connects to why leading terms dominate: far from the origin, or under repeated differencing, the lower-degree terms stop mattering and only the highest power survives. It also sets up a key limitation worth knowing before you trust a difference table: the whole method depends on equally spaced x-values. If the spacing is irregular, the differences never settle into a clean constant row, and the test for degree simply fails.