# Program to add two polynomials

Last Updated : 19 Mar, 2024

Given two polynomials represented by two arrays, write a function that adds given two polynomials.

Example:

Input:  A[] = {5, 0, 10, 6}
B[] = {1, 2, 4}
Output: sum[] = {6, 2, 14, 6}

The first input array represents "5 + 0x^1 + 10x^2 + 6x^3"
The second array represents "1 + 2x^1 + 4x^2"
And Output is "6 + 2x^1 + 14x^2 + 6x^3"

We strongly recommend minimizing your browser and try this yourself first.

Addition is simpler than multiplication of polynomials. We initialize the result as one of the two polynomials, then we traverse the other polynomial and add all terms to the result.

1) Create a sum array sum[] of size equal to maximum of 'm' and 'n'
2) Copy A[] to sum[].
3) Traverse array B[] and do following for every element B[i]
sum[i] = sum[i] + B[i]
4) Return sum[].

The following is the implementation of the above algorithm.

C++
// Simple C++ program to add two polynomials
#include <iostream>
using namespace std;

// A utility function to return maximum of two integers
int max(int m, int n) { return (m > n) ? m : n; }

// A[] represents coefficients of first polynomial
// B[] represents coefficients of second polynomial
// m and n are sizes of A[] and B[] respectively
int* add(int A[], int B[], int m, int n)
{
int size = max(m, n);
int* sum = new int[size];

// Initialize the product polynomial
for (int i = 0; i < m; i++)
sum[i] = A[i];

// Take every term of first polynomial
for (int i = 0; i < n; i++)
sum[i] += B[i];

return sum;
}

// A utility function to print a polynomial
void printPoly(int poly[], int n)
{
for (int i = 0; i < n; i++) {
cout << poly[i];
if (i != 0)
cout << "x^" << i;
if (i != n - 1)
cout << " + ";
}
}

// Driver program to test above functions
int main()
{
// The following array represents polynomial 5 + 10x^2 +
// 6x^3
int A[] = { 5, 0, 10, 6 };

// The following array represents polynomial 1 + 2x +
// 4x^2
int B[] = { 1, 2, 4 };
int m = sizeof(A) / sizeof(A[0]);
int n = sizeof(B) / sizeof(B[0]);

cout << "First polynomial is \n";
printPoly(A, m);
cout << "\nSecond polynomial is \n";
printPoly(B, n);

int* sum = add(A, B, m, n);
int size = max(m, n);

cout << "\nsum polynomial is \n";
printPoly(sum, size);

return 0;
}
Java
// Java program to add two polynomials

class GFG {

// A utility function to return maximum of two integers
static int max(int m, int n) {
return (m > n) ? m : n;
}

// A[] represents coefficients of first polynomial
// B[] represents coefficients of second polynomial
// m and n are sizes of A[] and B[] respectively
static int[] add(int A[], int B[], int m, int n) {
int size = max(m, n);
int sum[] = new int[size];

// Initialize the product polynomial
for (int i = 0; i < m; i++) {
sum[i] = A[i];
}

// Take ever term of first polynomial
for (int i = 0; i < n; i++) {
sum[i] += B[i];
}

return sum;
}

// A utility function to print a polynomial
static void printPoly(int poly[], int n) {
for (int i = 0; i < n; i++) {
System.out.print(poly[i]);
if (i != 0) {
System.out.print("x^" + i);
}
if (i != n - 1) {
System.out.print(" + ");
}
}
}

// Driver program to test above functions
public static void main(String[] args) {
// The following array represents polynomial 5 + 10x^2 + 6x^3
int A[] = {5, 0, 10, 6};

// The following array represents polynomial 1 + 2x + 4x^2
int B[] = {1, 2, 4};
int m = A.length;
int n = B.length;
System.out.println("First polynomial is");
printPoly(A, m);
System.out.println("\nSecond polynomial is");
printPoly(B, n);
int sum[] = add(A, B, m, n);
int size = max(m, n);
System.out.println("\nsum polynomial is");
printPoly(sum, size);

}
}
Python3
# Simple Python 3 program to add two
# polynomials

# A utility function to return maximum
# of two integers

# A[] represents coefficients of first polynomial
# B[] represents coefficients of second polynomial
# m and n are sizes of A[] and B[] respectively

size = max(m, n);
sum = [0 for i in range(size)]

# Initialize the product polynomial

for i in range(0, m, 1):
sum[i] = A[i]

# Take ever term of first polynomial
for i in range(n):
sum[i] += B[i]

return sum

# A utility function to print a polynomial
def printPoly(poly, n):
for i in range(n):
print(poly[i], end = "")
if (i != 0):
print("x^", i, end = "")
if (i != n - 1):
print(" + ", end = "")

# Driver Code
if __name__ == '__main__':

# The following array represents
# polynomial 5 + 10x^2 + 6x^3
A = [5, 0, 10, 6]

# The following array represents
# polynomial 1 + 2x + 4x^2
B = [1, 2, 4]
m = len(A)
n = len(B)

print("First polynomial is")
printPoly(A, m)
print("\n", end = "")
print("Second polynomial is")
printPoly(B, n)
print("\n", end = "")
sum = add(A, B, m, n)
size = max(m, n)

print("sum polynomial is")
printPoly(sum, size)

# This code is contributed by
# Sahil_Shelangia
C#
// C# program to add two polynomials
using System;
class GFG {

// A utility function to return maximum of two integers
static int max(int m, int n)
{
return (m > n) ? m : n;
}

// A[] represents coefficients of first polynomial
// B[] represents coefficients of second polynomial
// m and n are sizes of A[] and B[] respectively
static int[] add(int[] A, int[] B, int m, int n)
{
int size = max(m, n);
int[] sum = new int[size];

// Initialize the product polynomial
for (int i = 0; i < m; i++)
{
sum[i] = A[i];
}

// Take ever term of first polynomial
for (int i = 0; i < n; i++)
{
sum[i] += B[i];
}

return sum;
}

// A utility function to print a polynomial
static void printPoly(int[] poly, int n)
{
for (int i = 0; i < n; i++)
{
Console.Write(poly[i]);
if (i != 0)
{
Console.Write("x^" + i);
}
if (i != n - 1)
{
Console.Write(" + ");
}
}
}

// Driver code
public static void Main()
{
// The following array represents
// polynomial 5 + 10x^2 + 6x^3
int[] A = {5, 0, 10, 6};

// The following array represents
// polynomial 1 + 2x + 4x^2
int[] B = {1, 2, 4};
int m = A.Length;
int n = B.Length;
Console.WriteLine("First polynomial is");
printPoly(A, m);
Console.WriteLine("\nSecond polynomial is");
printPoly(B, n);
int[] sum = add(A, B, m, n);
int size = max(m, n);
Console.WriteLine("\nsum polynomial is");
printPoly(sum, size);

}
}

//This Code is Contributed
// by Mukul Singh
PHP
<?php
// Simple PHP program to add two polynomials

// A[] represents coefficients of first polynomial
// B[] represents coefficients of second polynomial
// m and n are sizes of A[] and B[] respectively
{
\$size = max(\$m, \$n);
\$sum = array_fill(0, \$size, 0);

// Initialize the product polynomial
for (\$i = 0; \$i < \$m; \$i++)
\$sum[\$i] = \$A[\$i];

// Take ever term of first polynomial
for (\$i = 0; \$i < \$n; \$i++)
\$sum[\$i] += \$B[\$i];

return \$sum;
}

// A utility function to print a polynomial
function printPoly(\$poly, \$n)
{
for (\$i = 0; \$i < \$n; \$i++)
{
echo \$poly[\$i];
if (\$i != 0)
echo "x^" . \$i;
if (\$i != \$n - 1)
echo " + ";
}
}

// Driver Code

// The following array represents
// polynomial 5 + 10x^2 + 6x^3
\$A = array(5, 0, 10, 6);

// The following array represents
// polynomial 1 + 2x + 4x^2
\$B = array(1, 2, 4);
\$m = count(\$A);
\$n = count(\$B);

echo "First polynomial is \n";
printPoly(\$A, \$m);
echo "\nSecond polynomial is \n";
printPoly(\$B, \$n);

\$sum = add(\$A, \$B, \$m, \$n);
\$size = max(\$m, \$n);

echo "\nsum polynomial is \n";
printPoly(\$sum, \$size);

// This code is contributed by chandan_jnu
?>
Javascript
<script>

// Simple JavaScript program to add two
// polynomials
// A utility function to return maximum
// of two integers

// A[] represents coefficients of first polynomial
// B[] represents coefficients of second polynomial
// m and n are sizes of A[] and B[] respectively
let size = Math.max(m, n);
var sum = [];
for (var i = 0; i < 10; i++) sum[i] = 0;
// Initialize the product polynomial
for(let i = 0;i<m;i++){
sum[i] = A[i];
}

// Take ever term of first polynomial
for (let i = 0;i<n;i++){
sum[i] += B[i];
}
return sum;
}

// A utility function to print a polynomial
function printPoly(poly, n){
let ans = '';
for(let i = 0;i<n;i++){
ans += poly[i];
if (i != 0){
ans +="x^ ";
ans +=i;
}
if (i != n - 1){
ans += " + ";
}
}
document.write(ans);
}
// Driver Code

// The following array represents
// polynomial 5 + 10x^2 + 6x^3
let A = [5, 0, 10, 6];
// The following array represents
// polynomial 1 + 2x + 4x^2
let B = [1, 2, 4];
let m = A.length;
let n = B.length;

document.write("First polynomial is" + "</br>");
printPoly(A, m);
document.write("</br>");
document.write("Second polynomial is"  + "</br>");
printPoly(B, n);
let sum = add(A, B, m, n);
let size = Math.max(m, n);
document.write("</br>");
document.write("sum polynomial is" + "</br>");
printPoly(sum, size);

</script>

Output:

First polynomial is
5 + 0x^1 + 10x^2 + 6x^3
Second polynomial is
1 + 2x^1 + 4x^2
Sum polynomial is
6 + 2x^1 + 14x^2 + 6x^3

Time complexity: O(m+n) where m and n are orders of two given polynomials.

Auxiliary Space: O(max(m, n))

C++
// Program to add two polynomials represented
#include <iostream>
using namespace std;

// Node class
class Node {

public:
int coeff, power;
Node* next;

// Constructor of Node
Node(int coeff, int power)
{
this->coeff = coeff;
this->power = power;
this->next = NULL;
}
};

{

// Checking if our list is empty
return;

// List contains elmements

}
<< " ";
}
else {
<< " ";
}
}

void insert(Node* head, int coeff, int power)
{
Node* new_node = new Node(coeff, power);
}
}

{
cout << "Linked List" << endl;
cout << " " << head->coeff << "x"
}
}

// Main function
int main()
{

Node* head = new Node(5, 2);
Node* head2 = new Node(6, 2);

cout << endl;

cout << endl << "Addition:" << endl;

return 0;
}
Java
// java code for the above approach

// Program to add two polynomials represented
import java.util.*;

// Node class
class Node {

public int coeff, power;
Node next;

// Constructor of Node
Node(int coeff, int power)
{
this.coeff = coeff;
this.power = power;
this.next = null;
}
}

public class Main {

{

// Checking if our list is empty
return;

// List contains elements

}
+ " ");
}
else {
+ " ");
}
}

public static void insert(Node head, int coeff, int power)
{
Node new_node = new Node(coeff, power);
}
}

{
System.out.print(" " + head.coeff + "x"
}
}

// Main function
public static void main(String[] args)
{

Node head = new Node(5, 2);
Node head2 = new Node(6, 2);

System.out.println();

}
}

// This code is contributed by Prince Kumar
Python3
# Program to add two polynomials represented
class Node:
def __init__(self, coeff, power):
self.coeff = coeff
self.power = power
self.next = None

return
else:

new_node = Node(coeff, power)

if __name__ == '__main__':
print()
C#
using System;

class Node
{
public int coeff, power;
public Node next;

// Constructor of Node
public Node(int coeff, int power)
{
this.coeff = coeff;
this.power = power;
this.next = null;
}
}

class Polynomial
{
{
// Checking if our list is empty
return;

// List contains elements
{
}
{
}
else
{
}
}

public static void Insert(Node head, int coeff, int power)
{
Node new_node = new Node(coeff, power);
{
}
}

{
{
}
}

public static void Main()
{
Node head = new Node(5, 2);
Node head2 = new Node(6, 2);

Console.WriteLine();

}
}
Javascript
// Program to add two polynomials represented

// Node class
class Node {
constructor(coeff, power) {
this.coeff = coeff;
this.power = power;
this.next = null;
}
}

// Checking if our list is empty
return;
}

// List contains elmements
} else {
}
}

const new_node = new Node(coeff, power);
}
}

}
}

// Main function
const head = new Node(5, 2);
const head2 = new Node(6, 2);

console.log();

Output
5x^2 4x^1
6x^2 4x^1
11x^2  8x^1

Time Complexity: O(m + n) where m and n are number of nodes in first and second lists respectively.
Auxiliary Space: O(m + n) where m and n are number of nodes in first and second lists respectively due to recursion.

### Implementation of a function that adds two polynomials represented as lists:

Approach

This implementation takes two arguments p1 and p2, which are lists representing the coefficients of two polynomials. The function returns a new list representing the sum of the two input polynomials.

The function first checks the lengths of the two input lists and pads the shorter list with zeros so that both lists have the same length. We then use the zip function to create pairs of corresponding coefficients from the two input lists, and the sum function to add the pairs together. The resulting sum is appended to a new list, which is returned at the end.

C++
#include <iostream>
#include <vector>

std::vector<int> add_polynomials(std::vector<int> p1, std::vector<int> p2) {
int len1 = p1.size();
int len2 = p2.size();

if (len1 < len2) {
p1.resize(len2, 0);
} else {
p2.resize(len1, 0);
}

std::vector<int> result(len1);
for (int i = 0; i < len1; i++) {
result[i] = p1[i] + p2[i];
}

return result;
}

int main() {
std::vector<int> p1 = {2, 0, 4, 6, 8};
std::vector<int> p2 = {0, 0, 1, 2};

for (int i = 0; i < result.size(); i++) {
std::cout << result[i] << " ";
}

return 0;
}
Python3
len1, len2 = len(p1), len(p2)
if len1 < len2:
p1 += [0] * (len2 - len1)
else:
p2 += [0] * (len1 - len2)
return [sum(x) for x in zip(p1, p2)]

p1 = [2, 0, 4, 6, 8]
p2 = [0, 0, 1, 2]
Java
import java.util.*;

public static List<Integer> addPolynomials(List<Integer> p1, List<Integer> p2) {
int len1 = p1.size();
int len2 = p2.size();

if (len1 < len2) {
for (int i = 0; i < len2 - len1; i++) {
}
} else {
for (int i = 0; i < len1 - len2; i++) {
}
}

List<Integer> result = new ArrayList<Integer>(len1);
for (int i = 0; i < len1; i++) {
}

return result;
}

public static void main(String[] args) {
List<Integer> p1 = new ArrayList<Integer>(Arrays.asList(2, 0, 4, 6, 8));
List<Integer> p2 = new ArrayList<Integer>(Arrays.asList(0, 0, 1, 2));

for (int i = 0; i < result.size(); i++) {
System.out.print(result.get(i) + " ");
}
}
}
C#
using System;
using System.Collections.Generic;
using System.Linq;

class Program {
static List<int> AddPolynomials(List<int> p1, List<int> p2) {
int len1 = p1.Count;
int len2 = p2.Count;

if (len1 < len2) {
} else {
}

List<int> result = new List<int>(len1);
for (int i = 0; i < len1; i++) {
}

return result;
}

static void Main(string[] args) {
List<int> p1 = new List<int> { 2, 0, 4, 6, 8 };
List<int> p2 = new List<int> { 0, 0, 1, 2 };

foreach (int coeff in result) {
Console.Write(coeff + " ");
}
}
}
Javascript
let len1 = p1.length;
let len2 = p2.length;

if (len1 < len2) {
p1 = p1.concat(new Array(len2 - len1).fill(0));
} else {
p2 = p2.concat(new Array(len1 - len2).fill(0));
}

let result = new Array(len1);
for (let i = 0; i < len1; i++) {
result[i] = p1[i] + p2[i];
}

return result;
}

let p1 = [2, 0, 4, 6, 8];
let p2 = [0, 0, 1, 2];

console.log(result.join(" "));

Output
[2, 0, 5, 8, 8]

time complexity: O(n), where n is the max of length of two polynomials

space complexity: O(n). where n is the max of length of two polynomials

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