Editing Synthetic division (section)

== Regular synthetic division ==
The first example is synthetic division with only a [[monic polynomial|monic]] linear denominator <math>x-a</math>.

:<math>\frac{x^3 - 12x^2 - 42}{x - 3}</math>

The [[numerator]] can be written as <math> p(x) = x^3 - 12x^2 + 0x - 42 </math>.

The zero of the denominator <math>g(x)</math> is <math>3</math>.

The coefficients of <math>p(x)</math> are arranged as follows, with the zero of <math>g(x)</math> on the left: 

:<math>\begin{array}{cc}
    \begin{array}{r} \\ 3 \\ \end{array}
    &
    \begin{array}{|rrrr} \ 
        1 & -12 & 0 & -42 \\
          &     &   &     \\
        \hline 
    \end{array}
\end{array}</math>

The {{font color|blue|first coefficient}} after the bar is "dropped" to the last row.
:<math>\begin{array}{cc}
    \begin{array}{r} \\ 3 \\ \\ \end{array}
    &
    \begin{array}{|rrrr}  
        \color{blue}1 & -12 & 0 & -42 \\
          &     &   &     \\
        \hline 
        \color{blue}1 &     &   &     \\
    \end{array}
\end{array}</math>

The {{font color|blue|dropped number}} is multiplied by the {{font color|grey|number}} before the bar and placed in the {{font color|brown|next column }}.
:<math>\begin{array}{cc}
    \begin{array}{r} \\ \color{grey}3 \\ \\ \end{array}
    &
    \begin{array}{|rrrr}  
        1 & -12 & 0 & -42 \\
          &   \color{brown}3 &   &     \\
        \hline 
        \color{blue}1 &     &   &     \\
    \end{array}
\end{array}</math>

An {{font color|green|addition}} is performed in the next column.
:<math>\begin{array}{cc}
    \begin{array}{c} \\ 3 \\ \\ \end{array}
    &
    \begin{array}{|rrrr}  
        1 & \color{green}-12 & 0 & -42 \\
          &   \color{green}3 &   &     \\
        \hline 
        1 &  \color{green}-9 &   &     \\
    \end{array}
\end{array}</math>

The previous two steps are repeated, and the following is obtained:
:<math>\begin{array}{cc}
    \begin{array}{c} \\ 3 \\ \\ \end{array}
    &
    \begin{array}{|rrrr}  
        1 & -12 &   0 & -42 \\
          &   3 & -27 & -81 \\
        \hline 
        1 & -9 & -27 & -123 
    \end{array}
\end{array}</math>

Here, the last term (-123) is the remainder while the rest correspond to the coefficients of the quotient. 
 
The terms are written with increasing degree from right to left beginning with degree zero for the remainder and the result.
:<math>	\begin{array}{rrr|r} 
    1x^2 &  -9x & -27 & -123 
\end{array}</math>

Hence the quotient and remainder are:

:<math>q(x) = x^2 - 9x - 27 
</math>
:<math>r(x) = -123</math>

===Evaluating polynomials by the remainder theorem===

The above form of synthetic division is useful in the context of the [[polynomial remainder theorem]] for evaluating [[univariate]] polynomials. To summarize, the value of <math>p(x)</math> at <math>a</math> is equal to the [[remainder]] of the division of <math>p(x)</math> by <math>x-a.</math>

The advantage of calculating the value this way is that it requires just over half as many multiplication steps as naive evaluation. An alternative evaluation strategy is [[Horner's method]].