Ruffini Rule

Ruffini's rule is an effective technique for dividing a polynomial by a bin of the form $x-r.$ ${\ displaystyle xr.}$ ${\ displaystyle x-r.}$ In 1804, it was described by Paolo Ruffini . ^[1] The Ruffini rule is a special case of synthetic division , when the divider is linear.

Content

1 Algorithm
2 Use
- 2.1 Division by the polynomial x - r
3 References
4 notes

Algorithm

The rule establishes a method for dividing a polynomial

P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}

{\ displaystyle P (x) = a_ {n} x ^ {n} + a_ {n-1} x ^ {n-1} + \ cdots + a_ {1} x + a_ {0}}

on bin

Q(x)=x-r

{\ displaystyle Q (x) = xr}

for private

R(x)=b_{n-1}x^{n-1}+b_{n-2}x^{n-2}+\cdots +b_{1}x+b_{0}

{\ displaystyle R (x) = b_ {n-1} x ^ {n-1} + b_ {n-2} x ^ {n-2} + \ cdots + b_ {1} x + b_ {0}}

;

In fact, the algorithm divides the column P ( x ) by Q ( x ).

In order to divide P ( x ) by Q ( x ) according to this algorithm, you need

Take the coefficients P ( x ) and write them in order. Then write r to the left, directly above the line:
${\begin{array}{c|c c c c c}&a_{n}&a_{n-1}&\dots &a_{1}&a_{0}\\r&&&&&\\\hline &&&&&\\&&&&&\\\end{array}}$ ${\ displaystyle {\ begin {array} {c | ccccc} & a_ {n} & a_ {n-1} & \ dots & a_ {1} & a_ {0} \\ r &&&&& \\\ hline &&&&&&\ {array}}}$
Move the leftmost coefficient ( a _n ) down, just below the line:
${\begin{array}{c|c c c c c}&a_{n}&a_{n-1}&\dots &a_{1}&a_{0}\\r&&&&&\\\hline &a_{n}&&&&\\&=b_{n-1}&&&&\\\end{array}}$ ${\ displaystyle {\ begin {array} {c | ccccc} & a_ {n} & a_ {n-1} & \ dots & a_ {1} & a_ {0} \\ r &&&&& \\\ hline & a_ {n} &&&& \\ & = b_ {n-1} &&&& \\\ end {array}}}$
Multiply the rightmost number under the line by r and write as follows above the line:
${\begin{array}{c|c c c c c}&a_{n}&a_{n-1}&\dots &a_{1}&a_{0}\\r&&b_{n-1}r&&&\\\hline &a_{n}&&&&\\&=b_{n-1}&&&&\\\end{array}}$ ${\ displaystyle {\ begin {array} {c | ccccc} & a_ {n} & a_ {n-1} & \ dots & a_ {1} & a_ {0} \\ r && b_ {n-1} r &&& \\\ hline & a_ { n} &&&& \\ & = b_ {n-1} &&&& \\\ end {array}}}$
Add two values located in one column:
${\begin{array}{c|c c c c c}&a_{n}&a_{n-1}&\dots &a_{1}&a_{0}\\r&&b_{n-1}r&&&\\\hline &a_{n}&a_{n-1}+(b_{n-1}r)&&&\\&=b_{n-1}&=b_{n-2}&&&\\\end{array}}$ ${\ displaystyle {\ begin {array} {c | ccccc} & a_ {n} & a_ {n-1} & \ dots & a_ {1} & a_ {0} \\ r && b_ {n-1} r &&& \\\ hline & a_ { n} & a_ {n-1} + (b_ {n-1} r) &&& \\ & = b_ {n-1} & = b_ {n-2} &&& \\\ end {array}}}$
Repeat steps 3 and 4 until there are numbers:
${\begin{array}{c|c c c c c}&a_{n}&a_{n-1}&\dots &a_{1}&a_{0}\\r&&b_{n-1}r&&&\\\hline &a_{n}&a_{n-1}+(b_{n-1}r)&\cdots &a_{1}+b_{1}r&a_{0}+b_{0}r\\&=b_{n-1}&=b_{n-2}&\cdots &=b_{0}&=s\\\end{array}}$ ${\ displaystyle {\ begin {array} {c | ccccc} & a_ {n} & a_ {n-1} & \ dots & a_ {1} & a_ {0} \\ r && b_ {n-1} r &&& \\\ hline & a_ { n} & a_ {n-1} + (b_ {n-1} r) & \ cdots & a_ {1} + b_ {1} r & a_ {0} + b_ {0} r \\ & = b_ {n-1} & = b_ {n-2} & \ cdots & = b_ {0} & = s \\\ end {array}}}$

The numbers b _i are the coefficients of the quotient ( R ( x )), whose degree is one less than the degree of P (x). The last s value obtained is the remainder . According to Bezout's theorem , this remainder is equal to P ( r ).

Usage

Division by polynomial x - r

A working example of division of polynomials according to the algorithm described above.

Let be:

P(x)=2x^{3}+3x^{2}-4,

{\ displaystyle P (x) = 2x ^ {3} + 3x ^ {2} -4,}

Q(x)=x+1.

{\ displaystyle Q (x) = x + 1.}

We want to find $P(x)/Q(x)$ ${\ displaystyle P (x) / Q (x)}$ using the Ruffini rule. The main problem is that $Q(x)$ ${\ displaystyle Q (x)}$ this is not a bin of the form $x-r,$ ${\ displaystyle xr,}$ but rather $x+r.$ ${\ displaystyle x + r.}$ We must rewrite it like this:

Q(x)=x+1=x-(-1).

{\ displaystyle Q (x) = x + 1 = x - (- 1).}

Now apply the algorithm:

1. We write out the coefficients and the number $r .$ ${\ displaystyle r.}$ Note that since $P(x)$ ${\ displaystyle P (x)}$ no coefficient $x^{1},$ ${\ displaystyle x ^ {1},}$ we write 0:

{\begin{array}{c|c c c c}&2&3&0&-4\\-1&&&&\\\hline &&&&\\&&&&\\\end{array}}

{\ displaystyle {\ begin {array} {c | cccc} & 2 & 3 & 0 & -4 \\ - 1 &&&& \\\ hline &&&& \\ &&&& \\\ end {array}}}

2. We lower the first coefficient:

{\begin{array}{c|c c c c}&2&3&0&-4\\-1&&&&\\\hline &2&&&\\\end{array}}

{\ displaystyle {\ begin {array} {c | cccc} & 2 & 3 & 0 & -4 \\ - 1 &&&& \\\ hline & 2 &&& \\\ end {array}}}

3. Multiply the last obtained value $r :$ ${\ displaystyle r:}$

{\begin{array}{c|c c c c}&2&3&0&-4\\-1&&-2&&\\\hline &2&&&\\\end{array}}

{\ displaystyle {\ begin {array} {c | cccc} & 2 & 3 & 0 & -4 \\ - 1 && - 2 && \\\ hline & 2 &&& \\\ end {array}}}

4. Add up the values:

{\begin{array}{c|c c c c}&2&3&0&-4\\-1&&-2&&\\\hline &2&1&&\\\end{array}}

{\ displaystyle {\ begin {array} {c | cccc} & 2 & 3 & 0 & -4 \\ - 1 && - 2 && \\\ hline & 2 & 1 && \\\ end {array}}}

5. Repeat steps 3 and 4:

{\begin{array}{c|c c c c}&2&3&0&-4\\-1&&-2&-1&1\\\hline &2&1&-1&-3\\\end{array}}

{\ displaystyle {\ begin {array} {c | cccc} & 2 & 3 & 0 & -4 \\ - 1 && - 2 & -1 & 1 \\\ hline & 2 & 1 & -1 & -3 \\\ end {array}}}

2,1,-1

{\ displaystyle 2.1, -1}

- the coefficients of the private,

-3

{\ displaystyle -3}

- the remainder.

So, since the original number = divisor × quotient + remainder , then

P(x)=Q(x)R(x)+s

{\ displaystyle P (x) = Q (x) R (x) + s}

where

R(x)=2x^{2}+x-1,\ s=-3;\quad \Rightarrow 2x^{3}+3x^{2}-4=(2x^{2}+x-1)(x+1)-3.

{\ displaystyle R (x) = 2x ^ {2} + x-1, \ s = -3; \ quad \ Rightarrow 2x ^ {3} + 3x ^ {2} -4 = (2x ^ {2} + x -1) (x + 1) -3.}

Links

Weisstein, Eric W. Ruffini Rule at Wolfram MathWorld .

Notes

↑ Cajory, Florian . Horner's method of approximation anticipated by Ruffini // Bulletin of the American Mathematical Society : journal. - 1911. - Vol. 17 , no. 8 . - P. 389-444 .

[1] Cajory, Florian . Horner's method of approximation anticipated by Ruffini // Bulletin of the American Mathematical Society : journal. - 1911. - Vol. 17 , no. 8 . - P. 389-444 .