Editing Ruffini's rule (section)

==Example==
Here is an example of polynomial division as described above.

Let:
<!-- The \,\! is to keep the formulae rendered as PNG instead of HTML to ensure consistency of representation. Please don't remove it.--> 
:<math>P(x)=2x^3+3x^2-4\,\!</math>
:<math>Q(x)=x+1.\,\!</math>

''P''(''x'') will be divided by ''Q''(''x'') using Ruffini's rule. The main problem is that ''Q''(''x'') is not a binomial of the form ''x'' − ''r'', but rather ''x'' + ''r''. ''Q''(''x'') must be rewritten as
:<math>Q(x)=x+1=x-(-1).\,\!</math>
Now the algorithm is applied:
<ol>
<li>Write down the coefficients and ''r''. Note that, as ''P''(''x'') didn't contain a coefficient for ''x'', 0 is written:
<pre>
     |     2     3     0  |  -4
     |                    |               
  -1 |                    |               
 ----|--------------------|-------
     |                    |               
     |                    |               
</pre>
</li>
<li>Pass the first coefficient down:
<pre>
     |     2     3     0  |  -4
     |                    |               
  -1 |                    |               
 ----|--------------------|-------
     |     2              |               
     |                    |               
</pre>
</li>
<li>Multiply the last obtained value by ''r'':
<pre>
     |     2     3     0  |  -4
     |                    |               
  -1 |          -2        |                
 ----|--------------------|-------
     |     2              |               
     |                    |               
</pre>
</li>
<li>Add the values:
<pre>
     |     2     3     0  |  -4
     |                    |
  -1 |          -2        |
 ----|--------------------|-------
     |     2     1        |
     |                    |               
</pre>
</li>
<li>Repeat steps 3 and 4 until it's finished:
<pre>
     |     2     3     0   | -4
     |                     |
  -1 |          -2    -1   |  1
 ----|----------------------------
     |     2     1    -1   | -3
     |{result coefficients}|{remainder}
</pre>
</li>
</ol>
<!-- The \,\! is to keep the formulae rendered as PNG instead of HTML to ensure consistency of representation. Please don't remove it.-->

So, if ''original number'' = ''divisor'' × ''quotient'' + ''remainder'', then
:<math>P(x)=Q(x)R(x)+s\,\!</math>, where

:<math>R(x) = 2x^2+x-1\,\!</math> and <math>s=-3; \quad \Rightarrow 2x^3+3x^2-4 = (2x^2+x-1)(x+1) - 3\!</math>