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{{draft task|Classic CS problems and programs}} {{Wikipedia|Synthetic division}}

:In algebra, [[wp:Synthetic division|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]].

## C#

{{trans|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;

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)
);
}
}
}

```

## Go

{{trans|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)
if coef := out[i]; coef.Sign() != 0 {
var a big.Rat
for j := 1; j < len(divisor); j++ {
}
}
}
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)
}
```

{{out}}

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

```

## J

Solving this the easy way:

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

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

```

## Java

{{trans|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;

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]]
```

## Kotlin

{{trans|Python}}

```// version 1.1.2

fun extendedSyntheticDivision(dividend: IntArray, divisor: IntArray): Pair<IntArray, IntArray> {
val out = dividend.copyOf()
val normalizer = divisor
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()}")
}
```

{{out}}

```
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

{{trans|Perl 6}}

```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;
\$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]);
```

{{out}}

```[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 6

{{trans|Python}} {{works with|Rakudo|2018.09}}

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

for ^(@numerator-\$end) -> \$i {
@result[\$i]    /= @denominator;
@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>]";
}
```

{{out}}

```[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

{{trans|Kotlin}}

```function extendedSyntheticDivision(sequence dividend, divisor)
sequence out = dividend
integer normalizer = divisor
integer 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 out[i+j-1] += -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, 0, 0, 0, -2},{1, 1, 1, 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
```

{{out}}

```
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}

```

## 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.x}}

```# -*- 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
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),
print " %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

{{trans|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))
```

{{out}}

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

## 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']'

```

{{out}}

```[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

{{Out}}Best seen running in your browser either by [https://scalafiddle.io/sf/59vpjcQ/0 ScalaFiddle (ES aka JavaScript, non JVM)] or [https://scastie.scala-lang.org/uUk8yRPnQdGdS1aAUFjhmA 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

{{trans|Python}}

```func extended_synthetic_division(dividend, divisor) {
var end = divisor.end
var out = dividend.clone
var normalizer = divisor

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))
```

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

## Tcl

{{trans|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
```

{{out}}

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

## zkl

{{trans|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;
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()));
```

{{out}}

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

```