Example+of+synthetic+step+by+step


 * [[image:http://upload.wikimedia.org/math/b/5/e/b5e8f030ff6ee0ebf91f8b20acbf3861.png caption="x-3overline{vert x^3 - 12x^2 + 0x - 42}"]] ||
 * x-3overline{vert x^3 - 12x^2 + 0x - 42} ||

We concern ourselves basically with the coefficients. We write
 * [[image:http://upload.wikimedia.org/math/a/a/4/aa43ff5a61faad8d7ebe9554cf94041d.png caption="begin{matrix}3 & | & 1 & -12& 0 & -42end{matrix}"]] ||
 * begin{matrix}3 & | & 1 & -12& 0 & -42end{matrix} ||

Note the change of sign from −3 to 3, the sign doesn't actually change, the value you are using (3 in this case) is the value that x would have to be in order for the polynomial to equal 0. Drop the first coefficient after the bar.
 * [[image:http://upload.wikimedia.org/math/9/2/8/928179e51659e48ea8cc82e28a1c6459.png caption="begin{matrix}3 & | & 1 & -12& 0 & -42 & | & & & & & | & 1 & & & end{matrix}"]] ||
 * begin{matrix}3 & | & 1 & -12& 0 & -42 & | & & & & & | & 1 & & & end{matrix} ||

Multiply the dropped number by the number before the bar, and place it in the next column.
 * [[image:http://upload.wikimedia.org/math/2/d/9/2d9f0c6db73caa1ecabc89ea8886d1a5.png caption="begin{matrix}3 & | & 1 & -12& 0 & -42 & | & & 3 & & & | & 1 & & & end{matrix}"]] ||
 * begin{matrix}3 & | & 1 & -12& 0 & -42 & | & & 3 & & & | & 1 & & & end{matrix} ||

Perform an addition in that column.
 * [[image:http://upload.wikimedia.org/math/5/c/5/5c51c04b06842604b4f109147f2f3926.png caption="begin{matrix}3 & | & 1 & -12& 0 & -42 & | & & 3 & & & | & 1 & -9 & & end{matrix}"]] ||
 * begin{matrix}3 & | & 1 & -12& 0 & -42 & | & & 3 & & & | & 1 & -9 & & end{matrix} ||

Repeat the previous two steps, the following is obtained
 * [[image:http://upload.wikimedia.org/math/8/4/1/84171c02b88d6072e227a1a604ed5e51.png caption="begin{matrix}3 & | & 1 & -12& 0 & -42 & | & & 3 & -27 & -81 & | & 1 & -9 & -27 & -123 end{matrix}"]] ||
 * begin{matrix}3 & | & 1 & -12& 0 & -42 & | & & 3 & -27 & -81 & | & 1 & -9 & -27 & -123 end{matrix} ||

All the numbers on the last row besides the farthest right correspond to coefficients in the quotient; the last number indicates a remainder. The terms are written with increasing degree from right to left, starting to the left of the remainder with degree 0. The result of our division is:
 * [[image:http://upload.wikimedia.org/math/a/1/8/a1885777cd4f1468dbe36d212276c2d7.png caption="frac{x^3 - 12x^2 - 42}{x - 3} = x^2 - 9x - 27 - frac{123}{x - 3}"]] ||
 * frac{x^3 - 12x^2 - 42}{x - 3} = x^2 - 9x - 27 - frac{123}{x - 3} ||