An [[algorithm]] for dividing a [[polynomial]] by linear [[factor]] The [[coefficient]] of the polynomial are arranged into an array, and then they, along with the zero of the linear factor, are used to generate the array of coefficients for the corresponding co-factor. If $p(x)=a_nx^n +a_{n-1}x^{n-1} +\ldots+a_2x^2+a_1x+a_0$ is divided by $x-a$ then synthetic division can be organized like this: ${\scriptsize \begin{array}{c|rrrrr} a & a_n & a_{n-1}&\ldots&a_1&a_0 \\& & a_n a&\ldots& a_na^{n-1} +a_{n-1}a^{n-2} +\ldots+a_2a^1 & a_na^{n} +a_{n-1}a^{n-1} +\ldots+a_2a^2+a_1a \\\hline & a_n & a_n a+a_{n-1}&\ldots&a_na^{n-1} +a_{n-1}a^{n-2} +\ldots+a_2a+a_1& \underset{\text{remainder}}{\underbrace{a_na^{n} +a_{n-1}a^{n-1} +\ldots+a_2a^2+a_1a+a_0}}\\\end{array} }$ ![[synthetic-division-and-remainder.svg]]