Horner's Scheme

Horner's scheme is a fast way to evaluate a polynomial $p(x_0)$ and divide it by the binomial $(x-x_0)$. Instead of computing powers, the polynomial is written as nested multiplication:

$a_nx^n+\dots+a_0=(\dots((a_n x+a_{n-1})x+a_{n-2})x+\dots)+a_0$

The running intermediate values are the coefficients of the quotient, and the last one is the remainder, which by Bézout's theorem equals $p(x_0)$. If the remainder is 0, then $x_0$ is a root of the polynomial (and $(x-x_0)$ is a factor of it).

The calculator shows the Horner table, the value $p(x_0)$ and the quotient. The method is handy for checking roots and lowering the degree of a polynomial.