Synthetic Division and the Division Algorithm

The division algorithm for polynomials states that, given $f(x)$ and a nonzero divisor $d(x)$, there exist unique polynomials $q(x)$ (quotient) and $r(x)$ (remainder), with $\deg r<\deg d$, such that $f(x)=d(x)q(x)+r(x)$. When $d(x)=x-a$ is linear, $r(x)$ is a constant, and by the Remainder Theorem (§4.1.1) that constant is $f(a)$ -- this is exactly what synthetic division computes.

For a numerical equation with no root immediately obvious, the trial-and-error method narrows the search: if the equation has integer coefficients and a rational root $p/q$ (in lowest terms), then $p$ must divide the constant term and $q$ must divide the leading coefficient. In particular, for a monic integer equation, every rational root is an integer dividing the constant term -- so only a short, finite list of candidates need be tried using synthetic division (equivalently, evaluating $f$ at each candidate). Once one root is confirmed (remainder $0$), the quotient is an equation of one degree lower, and the search continues on this depressed equation, which is usually far easier to finish (by the quadratic formula, once degree $2$ is reached, or by a further trial root). …