Ruffini's rule

To calculate the quotient of two polynomials the procedure used needs many intermediate calculations. A rule that can help us to simplify them is Ruffini's rule. This rule will only be valid when the divisor is a polynomial, such as $x-a$, with $a$ being a real number.

We will use an example to explain the methodology:

Do the division $\dfrac{p(x)}{q(x)}$, where $p(x)=x^4-3x^2+x+5$ and $q(x)=x+2$.

1. Complete and arrange the dividend polynomial.

Write the dividing polynomial as $x-a$, if necessary.

In our case:

$$p(x)=x^4+0x^3-3x^2+x+5$$

$$q(x)=x-(-2)$$

Notice that in this example the value of $a=-2$.

2. We write down the elements in a table like the following one.

	$1$	$0$	$-3$	$1$	$5$
$-2$

In the top row, we write the coefficients of the polynomial (arranged and completed!) $p(x)$.

In the left cell, we write the value of $a$.

3. We put the first coefficient in, and multiply it by the value of $a$. The result of which we write just under the second coefficient:

	$1$	$0$	$-3$	$1$	$5$
$-2$		$1\cdot(-2)=-2$
	$1$

4. We add up the second column and put the obtained result in, repeating the process until the last column:

	$1$	$0$	$-3$	$1$	$5$
$-2$		$1\cdot(-2)=-2$	$(-2)\cdot(-2)=4$	$1\cdot(-2)=-2$	$(-1)\cdot(-2)=2$
	$1$	$0+(-2)=-2$	$(-3)+4=1$	$1+(-2)=-1$	$5+2=7$

5. The digit on the bottom-right corner is the remainder. The other digits of the last row are the coefficients, arranged, for the polynomial quotient.

And so, in our case:

quotient: $x^3-2x^2+x-1$

remainder: $7$

As we can see, the relation od degrees is satisfied:

$3=$degree$(x^3-2x^2+x-1)=$degree$(x^4-3x^2+x+5)-$degree$(x+2)=4-1=3$

degree$(7)=0 < 1 =$degree$(x+2)$

Do the division $\dfrac{p(x)}{q(x)}$, where $p(x)=x^5+2x^4-3x^3+x^2-1$ and $q(x)=x-1$.

1. $p(x)=x^5+2x^4-3x^3+x^2+0x-1$

$$q(x)=x-1$$

$a=1$.

2. 	$1$	$2$	$-3$	$1$	$0$	$-1$
   $1$

3. 	$1$	$2$	$-3$	$1$	$0$	$-1$
   $1$		$1$
   	$1$	$3$

4. 	$1$	$2$	$-3$	$1$	$0$	$-1$
   $1$		$1$	$3$	$0$	$1$	$1$
   	$1$	$3$	$0$	$1$	$1$	$0$

quotient: $x^4+3x^3+x+1$

remainder: $0$

And it is satisfied that:

$4=$degree$(x^4+3x^3+x+1)=$degree$(x^5+2x^4-3x^3+x^2-1)-$

$-$degree$(x-1)=5-1=4$

degree$(0)=0 < 1 =$degree$(x-1)$