Ruffini's rule

Example

Do the division ${\displaystyle \frac{p(x)}{q(x)}}$, 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.

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:

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

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$(x+2)$

Example

Do the division ${\displaystyle \frac{p(x)}{q(x)}}$, 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)

3)

4)

5)

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$(x-1)$