Polynomials. Ruffini's Method.

You remember the mathematics of the school... You remember that the
numeric value of a polynomial should sometimes be calculated for a
well-known value of the variable "x". We see an example:

We know: f(x)=3*x^2 + 5*x + 2
We want to calculate f(4).
Then we calculate: 3*4^2 + 5*4 + 2 = 3*16 + 20 + 2 = 48 + 20 + 2 = 70
Therefore: f(4) = 70

For the method of Ruffini it is easier:

3 5 2
4 . 12 68
-------------
3 17 70

The method of Ruffini allows us to calculate f(a) for any value of "a"
in a polynomial well-known f(x), but that method is also useful to
make divisions of polynomials by (x-a). In our example:

(3*x^2 + 5*x + 2) / (x-4) = (3*x + 17) + (70 / (x-4))

If f(a) = 0 then are said that "a" it is a root of the polynomial, and
the rest of the division it would be zero.

Can you use the Ruffini's method in the soroban? Yes.

We will calculate again f(4), but using the abacus. The numeric
coefficients of f(x)=3*x^2 + 5*x + 2 are [3 5 2]. We will set those
coefficients using for each one of them three rods of the soroban. We
also set the 4 in any rod to the left:

MLKJIHGFEDCBA
...4..3..5..2 (J+4, G+3, D+5, A+2).

Multiply 3 on G by 4 on J, and add the product on ED:

MLKJIHGFEDCBA
...4..3.17..2 (4*3=12 -> E+1, D+2).

Now multiply 17 on ED by 4 on J and add the product on BA:

MLKJIHGFEDCBA
...4..3.17.70 (4*1=04 -> B+4; 4*7=28 -> B+5-2, A-2).

You can see the result of the calculation in the soroban.

What does it happen if some coefficient of the polynomial is a
negative number? I will only tell you that if the polynomial is, for
example, f(x)=4*x^2 - 8*x + 2 then I would set the numeric coefficients:

MLKJIHGFEDCBA
...4..41.8..2

The 1 on rod F means that the 8 of D are negative...

Enjoy using your soroban!

Saludos cordiales,

Fernando Tejón.