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This page uses content from Wikipedia. The original article was at Synthetic division. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)

In algebra, polynomial synthetic division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree in an efficient way using a trick involving clever manipulations of coefficients, which results in a lower time complexity than polynomial long division.

11l

Translation of: Python

F extended_synthetic_division(dividend, divisor)
   ‘Fast polynomial division by using Extended Synthetic Division. Also works with non-monic polynomials.’
   V out = copy(dividend)
   V normalizer = divisor[0]
   L(i) 0 .< dividend.len - (divisor.len - 1)
      out[i] /= normalizer
      V coef = out[i]
      I coef != 0
         L(j) 1 .< divisor.len
            out[i + j] += -divisor[j] * coef
   V separator = divisor.len - 1
   R (out[0 .< (len)-separator], out[(len)-separator..])

print(‘POLYNOMIAL SYNTHETIC DIVISION’)
V n = [1, -12, 0, -42]
V D = [1, -3]
print(‘  #. / #. =’.format(n, D), end' ‘ ’)
V (a, b) = extended_synthetic_division(n, D)
print(‘#. remainder #.’.format(a, b))

Output:

POLYNOMIAL SYNTHETIC DIVISION
  [1, -12, 0, -42] / [1, -3] = [1, -9, -27] remainder [-123]

C++

Translation of: Java

/*
 * C++ Polynomial Sythetic Division
 * GNU Compile example for filename <synthdiv.cpp>
 * g++ -std=c++11 -o synthdiv synthdiv.cpp
 */

#include <iostream>
#include <vector>
#include <string>
#include <cmath>

/*
 * frmtPolynomial method
 * Returns string for formatted 
 * polynomial from int vector of coefs.
 * String looks like ax^2 + bx + c, 
 * a, b, and c being the integer
 * coefs in the vector.
 */

std::string frmtPolynomial(std::vector<int> polynomial, bool remainder = false)
{
	std::string r = "";

	if (remainder) 
	{
		r = " r: " + std::to_string(polynomial.back());
		polynomial.pop_back();
	}

	std::string formatted = "";
	
	int degree = polynomial.size() - 1;
	int d = degree;

	for (int i : polynomial)
	{
		if (d < degree)
		{
			if (i >= 0) 
			{
				formatted += " + ";
			}
			else
			{
				formatted += " - ";
			}
		}

		formatted += std::to_string(abs(i));

		if (d > 1)
		{
			formatted += "x^" + std::to_string(d);
		}
		else if (d == 1)
		{
			formatted += "x";
		}

		d--;
	}

	return formatted;
}

/*
 * syntheticDiv Method
 * Performs Integer Polynomial Sythetic Division
 * on polynomials expressed as vectors of coefs.
 * Takes int vector param for dividend and 
 * divisor, and returns int vector quotient.
 */

std::vector<int> syntheticDiv(std::vector<int> dividend, std::vector<int> divisor)
{
	std::vector<int> quotient;
	quotient = dividend;

	int normalizer = divisor[0];
	
	for (int i = 0; i < dividend.size() - (divisor.size() - 1); i++)
	{
		quotient[i] /= normalizer;
		int coef = quotient[i];

		if (coef != 0) 
		{
			for (int j = 1; j < divisor.size(); j++)
			{
				quotient[i + j] += -divisor[j] * coef;
			}
        }

	}

	return quotient;
}

/*
 * Example of using the syntheticDiv method
 * and the frmtPolynomial method.
 * Assigns dividend and divisor polynomials:
 * dividend: 1x^3 - 12x^2 + 0x - 42
 * divisor: 1x - 3
 * Outputs both to cout using frmtPolynomial.
 * Printed polynomials look like above format.
 * Processes dividend and divisor in the 
 * syntheticDiv method, returns quotient.
 * Outputs quotient to cout using frmtPolynomial again.
 * quotient: 1x^2 - 9x - 27 r: -123
 */

int main(int argc, char **argv) 
{
	std::vector<int> dividend{ 1, -12, 0, -42};
	std::vector<int> divisor{ 1, -3};

	std::cout << frmtPolynomial(dividend) << "\n";
	std::cout << frmtPolynomial(divisor) << "\n";

	std::vector<int> quotient = syntheticDiv(dividend, divisor);

	std::cout << frmtPolynomial(quotient, true) << "\n";

}

C#

Translation of: Java

using System;
using System.Collections.Generic;
using System.Linq;

namespace SyntheticDivision
{
    class Program
    {
        static (List<int>,List<int>) extendedSyntheticDivision(List<int> dividend, List<int> divisor)
        {
            List<int> output = dividend.ToList();
            int normalizer = divisor[0];

            for (int i = 0; i < dividend.Count() - (divisor.Count() - 1); i++)
            {
                output[i] /= normalizer;

                int coef = output[i];
                if (coef != 0)
                {
                    for (int j = 1; j < divisor.Count(); j++)
                        output[i + j] += -divisor[j] * coef;
                }
            }

            int separator = output.Count() - (divisor.Count() - 1);

            return (
                output.GetRange(0, separator),
                output.GetRange(separator, output.Count() - separator)
            );
        }

        static void Main(string[] args)
        {
            List<int> N = new List<int>{ 1, -12, 0, -42 };
            List<int> D = new List<int> { 1, -3 };

            var (quotient, remainder) = extendedSyntheticDivision(N, D);
            Console.WriteLine("[ {0} ] / [ {1} ] = [ {2} ], remainder [ {3} ]" ,
                string.Join(",", N),
                string.Join(",", D),
                string.Join(",", quotient),
                string.Join(",", remainder)
            );
        }
    }
}

Delphi

Library: System.SysUtils

Library: Velthuis.BigRationals

Translation of: Go

Thanks Rudy Velthuis for the Velthuis.BigRationals library.

program Polynomial_synthetic_division;

{$APPTYPE CONSOLE}

uses
  System.SysUtils,
  Velthuis.BigRationals;

type
  TPollynomy = record
  public
    Terms: TArray<BigRational>;
    class operator Divide(a, b: TPollynomy): TArray<TPollynomy>;
    constructor Create(Terms: TArray<BigRational>);
    function ToString: string;
  end;

{ TPollynomy }

constructor TPollynomy.Create(Terms: TArray<BigRational>);
begin
  self.Terms := copy(Terms, 0, length(Terms));
end;

class operator TPollynomy.Divide(a, b: TPollynomy): TArray<TPollynomy>;
var
  q, r: TPollynomy;
begin
  var o: TArray<BigRational>;
  SetLength(o, length(a.Terms));
  for var i := 0 to High(a.Terms) do
    o[i] := BigRational.Create(a.Terms[i]);

  for var i := 0 to length(a.Terms) - length(b.Terms) do
  begin
    o[i] := BigRational.Create(o[i] div b.Terms[0]);
    var coef := BigRational.Create(o[i]);
    if coef.Sign <> 0 then
    begin
      var aa: BigRational := 0;
      for var j := 1 to High(b.Terms) do
      begin
        aa := (-b.Terms[j]) * coef;
        o[i + j] := o[i + j] + aa;
      end;
    end;
  end;
  var separator := length(o) - (length(b.Terms) - 1);

  q := TPollynomy.Create(copy(o, 0, separator));
  r := TPollynomy.Create(copy(o, separator, length(o)));

  result := [q, r];
end;

function TPollynomy.ToString: string;
begin
  Result := '[';
  for var e in Terms do
    Result := Result + e.ToString + ' ';
  Result := Result + ']';
end;

var
  p1, p2: TPollynomy;

begin
  p1 := TPollynomy.Create([BigRational.Create(1, 1), BigRational.Create(-12, 1),
    BigRational.Create(0, 1), BigRational.Create(-42, 1)]);

  p2 := TPollynomy.Create([BigRational.Create(1, 1), BigRational.Create(-3, 1)]);

  write(p1.ToString, ' / ', p2.ToString, ' = ');

  var result := p1 / p2;
  writeln(result[0].ToString, ' remainder ', result[1].ToString);
  readln;
end.

Output:

[1 -12 0 -42 ] / [1 -3 ] = [1 -9 -27 ] remainder [-123 ]

Go

Translation of: Python

package main

import (
    "fmt"
    "math/big"
)

func div(dividend, divisor []*big.Rat) (quotient, remainder []*big.Rat) {
    out := make([]*big.Rat, len(dividend))
    for i, c := range dividend {
        out[i] = new(big.Rat).Set(c)
    }
    for i := 0; i < len(dividend)-(len(divisor)-1); i++ {
        out[i].Quo(out[i], divisor[0])
        if coef := out[i]; coef.Sign() != 0 {
            var a big.Rat
            for j := 1; j < len(divisor); j++ {
                out[i+j].Add(out[i+j], a.Mul(a.Neg(divisor[j]), coef))
            }
        }
    }
    separator := len(out) - (len(divisor) - 1)
    return out[:separator], out[separator:]
}

func main() {
    N := []*big.Rat{
        big.NewRat(1, 1),
        big.NewRat(-12, 1),
        big.NewRat(0, 1),
        big.NewRat(-42, 1)}
    D := []*big.Rat{big.NewRat(1, 1), big.NewRat(-3, 1)}
    Q, R := div(N, D)
    fmt.Printf("%v / %v = %v remainder %v\n", N, D, Q, R)
}

Output:

[1/1 -12/1 0/1 -42/1] / [1/1 -3/1] = [1/1 -9/1 -27/1] remainder [-123/1]

Haskell

import Data.List

normalized :: (Eq a, Num a) => [a] -> [a]
normalized = dropWhile (== 0)

isZero :: (Eq a, Num a) => [a] -> Bool
isZero = null . normalized

shortDiv :: (Eq a, Fractional a) => [a] -> [a] -> ([a], [a])
shortDiv p1 p2
  | isZero p2 = error "zero divisor"
  | otherwise =
      let go 0 p = p 
          go i (h:t) = (h/a) : go (i-1) (zipWith (+) (map ((h/a) *) ker) t)
      in splitAt k $ go k p1
  where
    k = length p1 - length as
    a:as = normalized p2
    ker = negate <$> (as ++ repeat 0)

*Main> shortDiv [1,-12,0,-42] [1,1,-3]
([1.0,-13.0],[16.0,-81.0])

*Main> shortDiv [6,5,0,-7] [3,-2,-1]
([2.0,3.0],[8.0,-4.0])

For monic divisors it is possible to perform purely integral computations (without Fractional constraint):

isMonic :: (Eq a, Num a) => [a] -> Bool
isMonic = ([1] ==) . take 1 . normalized

shortDivMonic :: (Eq a, Num a) => [a] -> [a] -> ([a], [a])
shortDivMonic p1 p2
  | isZero p2 = error "zero divisor"
  | not (isMonic p2) = error "divisor is not monic"
  | otherwise = 
      let go 0 p = p 
          go i (h:t) = h : go (i-1) (zipWith (+) (map (h *) ker) t)
      in splitAt k $ go k p1
    where
      k = length p1 - length as
      _:as = normalized p2
      ker = negate <$> as ++ repeat 0

shortDivMonic [1,-12,0,-42] [1,1,-3 :: Int]
([1,-13],[16,-81])

J

Solving this the easy way:

   psd=: [:(}. ;{.) ([ (] -/@,:&}. (* {:)) ] , %&{.~)^:(>:@-~&#)~

Task example:

   (1, (-12), 0, -42) psd (1, -3)
┌────────┬────┐
│1 _9 _27│_123│
└────────┴────┘

Java

Translation of: Python

import java.util.Arrays;

public class Test {

    public static void main(String[] args) {
        int[] N = {1, -12, 0, -42};
        int[] D = {1, -3};

        System.out.printf("%s / %s = %s",
                Arrays.toString(N),
                Arrays.toString(D),
                Arrays.deepToString(extendedSyntheticDivision(N, D)));
    }

    static int[][] extendedSyntheticDivision(int[] dividend, int[] divisor) {
        int[] out = dividend.clone();
        int normalizer = divisor[0];

        for (int i = 0; i < dividend.length - (divisor.length - 1); i++) {
            out[i] /= normalizer;

            int coef = out[i];
            if (coef != 0) {
                for (int j = 1; j < divisor.length; j++)
                    out[i + j] += -divisor[j] * coef;
            }
        }

        int separator = out.length - (divisor.length - 1);

        return new int[][]{
            Arrays.copyOfRange(out, 0, separator),
            Arrays.copyOfRange(out, separator, out.length)
        };
    }
}

[1, -12, 0, -42] / [1, -3] = [[1, -9, -27], [-123]]

jq

Works with: jq

Works with gojq, the Go implementation of jq

Works with jaq, the Rust implementation of jq

In this entry, the polynomial of degree n, SIGMA c[i] * x^(n-i), is represented by the JSON array c.

# Solution: {"quotient", "remainder"}
def extendedSyntheticDivision($dividend; $divisor):
  { out: $dividend,
    normalizer: $divisor[0],
    separator: (($dividend|length) - ($divisor|length) + 1) }
  | reduce range(0; .separator) as $i (.;
      .out[$i] = ((.out[$i] / .normalizer)|trunc)
      | .out[$i] as $coef
      | if $coef != 0
        then reduce range(1; $divisor|length) as $j (.;
          .out[$i + $j] -=  $divisor[$j] * $coef )
        else .
        end )
  | {quotient: .out[0:.separator], remainder: .out[.separator:]} ;

def task($n; $d):
  def r: if length==1 then first else . end;
  extendedSyntheticDivision($n; $d)
  | "\($n) / \($d)  =  \(.quotient), remainder \(.remainder|r)" ;

task([1, -12, 0, -42]; [1, -3]),
task([1, 0, 0, 0, -2];[1, 1, 1, 1])

Output:

[1,-12,0,-42] / [1,-3]  =  [1,-9,-27], remainder -123
[1,0,0,0,-2] / [1,1,1,1]  =  [1,-1], remainder [0,0,-1]

Julia

Translation of: Perl

function divrem(dividend::Vector, divisor::Vector)
    result = copy(dividend)
    quotientlen = length(divisor) - 1
    for i in 1:length(dividend)-quotientlen
        if result[i] != 0
            result[i] /= divisor[1]
            for j in 1:quotientlen
                result[i + j] -= divisor[j + 1] * result[i]
            end
        end
    end
    return result[1:end-quotientlen], result[end-quotientlen+1:end]
end

testpolys = [([1, -12, 0, -42], [1, -3]), ([1, 0, 0, 0, -2], [1, 1, 1, 1])]

for (n, d) in testpolys
    quotient, remainder = divrem(n, d)
    println("[$n] / [$d] = [$quotient] with remainder [$remainder]")
end

Output:

[[1, -12, 0, -42]] / [[1, -3]] = [[1, -9, -27]] with remainder [[-123]]
[[1, 0, 0, 0, -2]] / [[1, 1, 1, 1]] = [[1, -1]] with remainder [[0, 0, -1]]

Kotlin

Translation of: Python

// version 1.1.2

fun extendedSyntheticDivision(dividend: IntArray, divisor: IntArray): Pair<IntArray, IntArray> {
    val out = dividend.copyOf()
    val normalizer = divisor[0]
    val separator = dividend.size - divisor.size + 1
    for (i in 0 until separator) {
        out[i] /= normalizer
        val coef = out[i]
        if (coef != 0) { 
            for (j in 1 until divisor.size) out[i + j] += -divisor[j] * coef
        }
    }
    return out.copyOfRange(0, separator) to out.copyOfRange(separator, out.size) 
}

fun main(args: Array<String>) {
    println("POLYNOMIAL SYNTHETIC DIVISION")
    val n = intArrayOf(1, -12, 0, -42)
    val d = intArrayOf(1, -3)
    val (q, r) = extendedSyntheticDivision(n, d)
    print("${n.contentToString()} / ${d.contentToString()}  =  ")
    println("${q.contentToString()}, remainder ${r.contentToString()}")
    println()
    val n2 = intArrayOf(1, 0, 0, 0, -2)
    val d2 = intArrayOf(1, 1, 1, 1)
    val (q2, r2) = extendedSyntheticDivision(n2, d2)
    print("${n2.contentToString()} / ${d2.contentToString()}  =  ")
    println("${q2.contentToString()}, remainder ${r2.contentToString()}")
}

Output:

POLYNOMIAL SYNTHETIC DIVISION
[1, -12, 0, -42] / [1, -3]  =  [1, -9, -27], remainder [-123]

[1, 0, 0, 0, -2] / [1, 1, 1, 1]  =  [1, -1], remainder [0, 0, -1]

Mathematica / Wolfram Language

MakePolynomial[l_List, x_] := FromCoefficientRules[Thread[List /@ Range[Length[l] - 1, 0, -1] -> l], {x}]
num = MakePolynomial[{1, -12, 0, -42}, x];
den = MakePolynomial[{1, -3}, x];
PolynomialQuotient[num, den, x]
PolynomialRemainder[num, den, x]

Output:

-27 - 9 x + x^2
-123

Nim

Translation of: Kotlin

import strformat

type Polynomial = seq[int]

func `$`(p: Polynomial): string = system.`$`(p)[1..^1]

func extendedSyntheticDivision(dividend, divisor: Polynomial): tuple[q, r: Polynomial] =
    var res = dividend
    let normalizer = divisor[0]
    let separator = dividend.len - divisor.len
    for i in 0..separator:
      res[i] = res[i] div normalizer
      let coef = res[i]
      if coef != 0:
        for j in 1..divisor.high:
          res[i + j] += -divisor[j] * coef
    result = (res[0..separator], res[(separator+1)..^1])

when isMainModule:
  echo "Polynomial synthetic division"
  let n1 = @[1, -12, 0, -42]
  let d1 = @[1, -3]
  let (q1, r1) = extendedSyntheticDivision(n1, d1)
  echo &"{n1} / {d1}  =  {q1}, remainder {r1}"
  let n2 = @[1, 0, 0, 0, -2]
  let d2 = @[1, 1, 1, 1]
  let (q2, r2) = extendedSyntheticDivision(n2, d2)
  echo &"{n2} / {d2}  =  {q2}, remainder {r2}"

Output:

Polynomial synthetic division
[1, -12, 0, -42] / [1, -3]  =  [1, -9, -27], remainder [-123]
[1, 0, 0, 0, -2] / [1, 1, 1, 1]  =  [1, -1], remainder [0, 0, -1]

Perl

Translation of: Raku

sub synthetic_division {
    my($numerator,$denominator) = @_;
    my @result = @$numerator;
    my $end    = @$denominator-1;

    for my $i (0 .. @$numerator-($end+1)) {
        next unless $result[$i];
        $result[$i]    /= @$denominator[0];
        $result[$i+$_] -= @$denominator[$_] * $result[$i] for 1 .. $end;
    }

    return join(' ', @result[0 .. @result-($end+1)]), join(' ', @result[-$end .. -1]);
}

sub poly_divide {
    *n = shift; *d = shift;
    my($quotient,$remainder)= synthetic_division( \@n, \@d );
    "[@n] / [@d] = [$quotient], remainder [$remainder]\n";
}

print poly_divide([1, -12, 0, -42], [1, -3]);
print poly_divide([1, 0, 0, 0, -2], [1, 1, 1, 1]);

Output:

[1 -12 0 -42] / [1 -3] = [1 -9 -27], remainder [-123]
[1 0 0 0 -2] / [1 1 1 1] = [1 -1], remainder [0 0 -1]

Phix

Translation of: Kotlin

with javascript_semantics
function extendedSyntheticDivision(sequence dividend, divisor)
    sequence out = deep_copy(dividend)
    integer normalizer = divisor[1],
            separator = length(dividend) - length(divisor) + 1
    for i=1 to separator do
        out[i] /= normalizer
        integer coef = out[i]
        if (coef != 0) then
            for j=2 to length(divisor) do
                integer odx = i+j-1
                out[odx] += -divisor[j] * coef
            end for
        end if
    end for
    return {out[1..separator],out[separator+1..$]}
end function
 
constant tests = {{{1, -12, 0, -42},{1, -3}},
                  {{1, -12, 0, -42},{1, 1, -3}},
                  {{1, 0, 0, 0, -2},{1, 1, 1, 1}},
                  {{6, 5, 0, -7},{3, -2, -1}}}
 
printf(1,"Polynomial synthetic division\n")
for t=1 to length(tests) do
    sequence {n,d} = tests[t],
             {q,r} = extendedSyntheticDivision(n, d)
    printf(1,"%v / %v  =  %v, remainder %v\n",{n,d,q,r})
end for

Output:

Polynomial synthetic division
{1,-12,0,-42} / {1,-3}  =  {1,-9,-27}, remainder {-123}
{1,-12,0,-42} / {1,1,-3}  =  {1,-13}, remainder {16,-81}
{1,0,0,0,-2} / {1,1,1,1}  =  {1,-1}, remainder {0,0,-1}
{6,5,0,-7} / {3,-2,-1}  =  {2,3}, remainder {8,-4}

Python

Here is an extended synthetic division algorithm, which means that it supports a divisor polynomial (instead of just a monomial/binomial). It also supports non-monic polynomials (polynomials which first coefficient is different than 1). Polynomials are represented by lists of coefficients with decreasing degree (left-most is the major degree , right-most is the constant).

Works with: Python 2.7

Works with: Python 3.x

from __future__ import print_function
from __future__ import division

#!/usr/bin/python
# coding=UTF-8

def extended_synthetic_division(dividend, divisor):
    '''Fast polynomial division by using Extended Synthetic Division. Also works with non-monic polynomials.'''
    # dividend and divisor are both polynomials, which are here simply lists of coefficients. Eg: x^2 + 3x + 5 will be represented as [1, 3, 5]

    out = list(dividend) # Copy the dividend
    normalizer = divisor[0]
    for i in xrange(len(dividend)-(len(divisor)-1)):
        out[i] /= normalizer # for general polynomial division (when polynomials are non-monic),
                                 # we need to normalize by dividing the coefficient with the divisor's first coefficient
        coef = out[i]
        if coef != 0: # useless to multiply if coef is 0
            for j in xrange(1, len(divisor)): # in synthetic division, we always skip the first coefficient of the divisor,
                                              # because it's only used to normalize the dividend coefficients
                out[i + j] += -divisor[j] * coef

    # The resulting out contains both the quotient and the remainder, the remainder being the size of the divisor (the remainder
    # has necessarily the same degree as the divisor since it's what we couldn't divide from the dividend), so we compute the index
    # where this separation is, and return the quotient and remainder.
    separator = -(len(divisor)-1)
    return out[:separator], out[separator:] # return quotient, remainder.

if __name__ == '__main__':
    print("POLYNOMIAL SYNTHETIC DIVISION")
    N = [1, -12, 0, -42]
    D = [1, -3]
    print("  %s / %s  =" % (N,D), " %s remainder %s" % extended_synthetic_division(N, D))

Sample output:

POLYNOMIAL SYNTHETIC DIVISION
  [1, -12, 0, -42] / [1, -3]  =  [1, -9, -27] remainder [-123]

Racket

Translation of: Python

#lang racket/base
(require racket/list)
;; dividend and divisor are both polynomials, which are here simply lists of coefficients.
;; Eg: x^2 + 3x + 5 will be represented as (list 1 3 5)
(define (extended-synthetic-division dividend divisor)
  (define out (list->vector dividend)) ; Copy the dividend
  ;; for general polynomial division (when polynomials are non-monic), we need to normalize by
  ;; dividing the coefficient with the divisor's first coefficient
  (define normaliser (car divisor))
  (define divisor-length (length divisor)) ; } we use these often enough
  (define out-length (vector-length out))  ; }
  
  (for ((i (in-range 0 (- out-length divisor-length -1))))
    (vector-set! out i (quotient (vector-ref out i) normaliser))
    (define coef (vector-ref out i))
    (unless (zero? coef) ; useless to multiply if coef is 0
      (for ((i+j (in-range (+ i 1)                ; in synthetic division, we always skip the first
                           (+ i divisor-length))) ; coefficient of the divisior, because it's
            (divisor_j (in-list (cdr divisor))))  ;  only used to normalize the dividend coefficients
        (vector-set! out i+j (+ (vector-ref out i+j) (* coef divisor_j -1))))))
  ;; The resulting out contains both the quotient and the remainder, the remainder being the size of
  ;; the divisor (the remainder has necessarily the same degree as the divisor since it's what we
  ;; couldn't divide from the dividend), so we compute the index where this separation is, and return
  ;; the quotient and remainder.

  ;; return quotient, remainder (conveniently like quotient/remainder)
  (split-at (vector->list out) (- out-length (sub1 divisor-length))))

(module+ main
  (displayln "POLYNOMIAL SYNTHETIC DIVISION")
  (define N '(1 -12 0 -42))
  (define D '(1 -3))
  (define-values (Q R) (extended-synthetic-division N D))
  (printf "~a / ~a = ~a remainder ~a~%" N D Q R))

Output:

POLYNOMIAL SYNTHETIC DIVISION
(1 -12 0 -42) / (1 -3) = (1 -9 -27) remainder (-123)

Raku

(formerly Perl 6)

Translation of: Python

Works with: Rakudo version 2018.09

sub synthetic-division ( @numerator, @denominator ) {
    my @result = @numerator;
    my $end    = @denominator.end;

    for ^(@numerator-$end) -> $i {
        @result[$i]    /= @denominator[0];
        @result[$i+$_] -= @denominator[$_] * @result[$i] for 1..$end;
    }

    'quotient' => @result[0 ..^ *-$end],
    'remainder' => @result[*-$end .. *];
}

my @tests =
[1, -12, 0, -42], [1, -3],
[1, 0, 0, 0, -2], [1, 1, 1, 1];

for @tests -> @n, @d {
    my %result = synthetic-division( @n, @d );
    say "[{@n}] / [{@d}] = [%result<quotient>], remainder [%result<remainder>]";
}

Output:

[1 -12 0 -42] / [1 -3] = [1 -9 -27], remainder [-123] 
[1 0 0 0 -2] / [1 1 1 1] = [1 -1], remainder [0 0 -1]

REXX

/* REXX Polynomial Division                */
/* extended to support order of divisor >1 */
call set_dd '1 0 0 0 -1'
Call set_dr '1 1 1 1'
Call set_dd '1 -12 0 -42'
Call set_dr '1 -3'
q.0=0
Say list_dd '/' list_dr
do While dd.0>=dr.0
  q=dd.1/dr.1
  Do j=1 To dr.0
    dd.j=dd.j-q*dr.j
    End
  Call set_q q
  Call shift_dd
  End
say 'Quotient:' mk_list_q() 'Remainder:' mk_list_dd()
Exit

set_dd:
Parse Arg list
list_dd='['
Do i=1 To words(list)
  dd.i=word(list,i)
  list_dd=list_dd||dd.i','
  End
dd.0=i-1
list_dd=left(list_dd,length(list_dd)-1)']'
Return

set_dr:
Parse Arg list
list_dr='['
Do i=1 To words(list)
  dr.i=word(list,i)
  list_dr=list_dr||dr.i','
  End
dr.0=i-1
list_dr=left(list_dr,length(list_dr)-1)']'
Return

set_q:
z=q.0+1
q.z=arg(1)
q.0=z
Return

shift_dd:
Do i=2 To dd.0
  ia=i-1
  dd.ia=dd.i
  End
dd.0=dd.0-1
Return

mk_list_q:
list='['q.1''
Do i=2 To q.0
  list=list','q.i
  End
Return list']'

mk_list_dd:
list='['dd.1''
Do i=2 To dd.0
  list=list','dd.i
  End
Return list']'

Output:

[1,-12,0,-42] / [1,-3]
Quotient: [1,-9,-27] Remainder: -123

[1,0,0,0,-2] / [1,1,1,1]
Quotient: [1,-1] Remainder: [0,0,-1]

Scala

Java Interoperability

Output:

Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).

import java.util

object PolynomialSyntheticDivision extends App {

  val N: Array[Int] = Array(1, -12, 0, -42)
  val D: Array[Int] = Array(1, -3)

  def extendedSyntheticDivision(dividend: Array[Int],
                                divisor: Array[Int]): Array[Array[Int]] = {
    val out = dividend.clone
    val normalizer = divisor(0)

    for (i <- 0 until dividend.length - (divisor.length - 1)) {
      out(i) /= normalizer
      val coef = out(i)
      if (coef != 0)
        for (j <- 1 until divisor.length) out(i + j) += -divisor(j) * coef
    }
    val separator = out.length - (divisor.length - 1)
    Array[Array[Int]](util.Arrays.copyOfRange(out, 0, separator),
      util.Arrays.copyOfRange(out, separator, out.length))
  }

  println(f"${util.Arrays.toString(N)}%s / ${util.Arrays.toString(D)}%s = ${
    util.Arrays
      .deepToString(extendedSyntheticDivision(N, D).asInstanceOf[Array[AnyRef]])
  }%s")

}

Sidef

Translation of: Python

func extended_synthetic_division(dividend, divisor) {
    var end = divisor.end
    var out = dividend.clone
    var normalizer = divisor[0]

    for i in ^(dividend.len - end) {
        out[i] /= normalizer
        var coef = out[i]
        if (coef != 0) {
            for j in (1 .. end) {
                out[i+j] += -(divisor[j] * coef)
            }
        }
    }

    var remainder = out.splice(-end)
    var quotient = out

    return(quotient, remainder)
}

var (n, d) = ([1, -12, 0, -42], [1, -3])
print("  %s / %s =" % (n, d))
print(" %s remainder %s\n" % extended_synthetic_division(n, d))

Output:

 [1, -12, 0, -42] / [1, -3] = [1, -9, -27] remainder [-123]

Tcl

Translation of: Python

This uses a common utility proc range, and a less common one called lincr, which increments elements of lists. The routine for polynomial division is placed in a namespace ensemble, such that it can be conveniently shared with other commands for polynomial arithmetic (eg polynomial multiply).

#   range ?start? end+1
# start defaults to 0:  [range 5] = {0 1 2 3 4}
proc range {a {b ""}} {
    if {$b eq ""} {
        set b $a
        set a 0
    }
    for {set r {}} {$a<$b} {incr a} {
        lappend r $a
    }
    return $r
}

#   lincr list idx ?...? increment
# By analogy with [lset] and [incr]:
# Adds incr to the item at [lindex list idx ?...?].  incr may be a float.
proc lincr {_ls args} {
    upvar 1 $_ls ls
    set incr [lindex $args end]
    set idxs [lrange $args 0 end-1]
    lset ls {*}$idxs [expr {$incr + [lindex $ls {*}$idxs]}]
}

namespace eval polynomial {
    # polynomial division, returns [list $dividend $remainder]
    proc divide {top btm} {
        set out $top
        set norm [lindex $btm 0]
        foreach i [range [expr {[llength $top] - [llength $btm] + 1}]] {
            lset out $i [set coef [expr {[lindex $out $i] * 1.0 / $norm}]]
            if {$coef != 0} {
                foreach j [range 1 [llength $btm]] {
                    lincr out [expr {$i+$j}] [expr {-[lindex $btm $j] * $coef}]
                }
            }
        }
        set terms [expr {[llength $btm]-1}]
        list [lrange $out 0 end-$terms] [lrange $out end-[incr terms -1] end]
    }
    namespace export *
    namespace ensemble create
}

proc test {} {
    set top {1 -12 0 -42}
    set btm {1 -3}
    set div [polynomial divide $top $btm]
    puts "$top / $btm = $div"
}
test

Output:

1 -12 0 -42 / 1 -3 = {1.0 -9.0 -27.0} -123.0

Wren

Translation of: Kotlin

Library: Wren-dynamic

import "./dynamic" for Tuple

var Solution = Tuple.create("Solution", ["quotient", "remainder"])

var extendedSyntheticDivision = Fn.new { |dividend, divisor|
    var out = dividend.toList
    var normalizer = divisor[0]
    var separator = dividend.count - divisor.count + 1
    for (i in 0...separator) {
        out[i] = (out[i] / normalizer).truncate
        var coef = out[i]
        if (coef != 0) {
            for (j in 1...divisor.count) out[i + j] = out[i + j] - divisor[j] * coef
        }
    }
    return Solution.new(out[0...separator], out[separator..-1])
}

System.print("POLYNOMIAL SYNTHETIC DIVISION")
var n = [1, -12, 0, -42]
var d = [1, -3]
var sol = extendedSyntheticDivision.call(n, d)
System.write("%(n) / %(d)  =  ")
System.print("%(sol.quotient), remainder %(sol.remainder)")
System.print()
var n2 = [1, 0, 0, 0, -2]
var d2 = [1, 1, 1, 1]
var sol2 = extendedSyntheticDivision.call(n2, d2)
System.write("%(n2) / %(d2)  =  ")
System.print("%(sol2.quotient), remainder %(sol2.remainder)")

Output:

POLYNOMIAL SYNTHETIC DIVISION
[1, -12, 0, -42] / [1, -3]  =  [1, -9, -27], remainder [-123]

[1, 0, 0, 0, -2] / [1, 1, 1, 1]  =  [1, -1], remainder [0, 0, -1]

zkl

Translation of: Python

fcn extended_synthetic_division(dividend, divisor){
# Fast polynomial division by using Extended Synthetic Division. Also works with non-monic polynomials.
# dividend and divisor are both polynomials, which are here simply lists of coefficients. Eg: x^2 + 3x + 5 will be represented as [1, 3, 5]
   out,normalizer:=dividend.copy(), divisor[0];
   foreach i in (dividend.len() - (divisor.len() - 1)){
      out[i] /= normalizer; # for general polynomial division (when polynomials are non-monic),
                            # we need to normalize by dividing the coefficient with the divisor's first coefficient
      coef := out[i];
      if(coef != 0){  # useless to multiply if coef is 0
	 foreach j in ([1..divisor.len() - 1]){ # in synthetic division, we always skip the first coefficient of the divisior,
	    out[i + j] += -divisor[j] * coef;   # because it's only used to normalize the dividend coefficients
	 }
      }
   }

    # out contains the quotient and remainder, the remainder being the size of the divisor (the remainder
    # has necessarily the same degree as the divisor since it's what we couldn't divide from the dividend), so we compute the index
    # where this separation is, and return the quotient and remainder.
   separator := -(divisor.len() - 1);
   return(out[0,separator], out[separator,*]) # return quotient, remainder.
}

println("POLYNOMIAL SYNTHETIC DIVISION");
N,D := T(1, -12, 0, -42), T(1, -3);
print("  %s / %s =".fmt(N,D));
println(" %s remainder %s".fmt(extended_synthetic_division(N,D).xplode()));

Output:

POLYNOMIAL SYNTHETIC DIVISION
  L(1,-12,0,-42) / L(1,-3) = L(1,-9,-27) remainder L(-123)