# Minimize the sum of roots of a given polynomial in C++

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We are given with an array of integer elements representing the coefficient values of a polynomial. The size of an array will be ‘n’ i.e. number of elements in an array. The degree of polynomial always starts with n-1 as there will be one constant value at the end of the polynomial series. The task is to replace the coefficient with other polynomials in such a manner that sum of roots will be minimized.

Let us see various input output scenarios for this -

In − int arr[] = { 2, -1, 4, 9, -1, 10, -5}

Out − Minimize the sum of roots of a given polynomial is: -1 -5 2 4 9 -1 10

Explanation − we are given with an integer array containing 7 elements i.e. polynomial power will be 6. So, we will achieve the minimum output as-: -1 * x^6 - 5 * x^5 + 2 * x^4 + 4 * x^3 + 9 * x^2 - 1 * x^1 + 10 which will result in a minimum sum of roots as -5 and 1.

In − int arr[] = {3, -2, -1, 4}

Out − Minimize the sum of roots of a given polynomial is: -1 -2 3 4

Explanation − we are given with an integer array containing 7 elements i.e. polynomial power will be 6. So, we will achieve the minimum output as-: -1 * x^3 - 2 * x^2 + 3 * x^1 + 4 which will result in a minimum sum of roots as -1.

## Approach used in the below program is as follows

• Input an array of integer elements and calculate the size of an array. Pass the data to the function for further processing.

• Inside the function Minimize_root(arr, size)

• Declare variables of type vector of integer type as vec_1, vec_2, vec_3.

• Start loop FOR from i to 0 till the size of an array. Inside the loop, check IF arr[i] > 0 then push_back i to vec_2. Else IF, arr[i] < 0 then puch_back i to vec_3.

• Calculate the size of vec_2 and vec_3.

• Check IF vec_2_size >= 2 AND IF vec_3_size>= 2 then start loop FOR from i to 0 till the size of vec_2 and inside the loop, check IFarr[vec_2[i]] greater than max_val then set temp to vec_2[i] and max_val to arr[temp].

• Start loop FOR from i to 0 till the size of a vec_2. Inside the loop, check IF arr[vec_2[i]] less than min_val than check IF vec_2[i] not equals to temp then set temp_2 to vec_2[i] and min_val to arr[temp_2]

• Start loop FOR from i to 0 till the size of a vec_3. Inside the loop, check IF abs(arr[vec_3[i]]) greater than N_max then set N_temp to vec_3[i] and N_max to abs(arr[N_temp])

• Start loop FOR from i to 0 till vec_3 size and check IF abs(arr[vec_3[i]]) less than N_min than check IFvec_3[i] not equals to N_temp then set N_temp_2 to vec_3[i] and N_min = abs(arr[N_temp_2])

• Check IF vec_2_data less than vec_3_data then inside the FOR loop from i to 0 till the size of an array, check IF i not equals to temp_2 AND i not equals to temp then push_back arr[i] to vec_1. ELSE, inside the FOR loop from i to 0 till the size of an array, check IF i not equals to N_temp_2 AND i not equals to N_temp then push_back arr[i] to vec_1.

• Start loop FOR to traverse the vec_1 and keep printing the vec_1[i] as a result.

• ELSE IF, check vec_2_size >= 2 then start loop FOR from i to 0 till the size of a vec_2. Check IF arr[vec_2[i]] greater than max_val then set temp to vec_2[i] and max_val to arr[temp].

• Start loop FOR from i to 0 till i less than size of a vec_2. Inside the loop, check IF arr[vec_2[i]] less than min_val than check IFvec_2[i] not equals to temp than set temp_2 to vec_2[i] and min_val to arr[temp_2].

• Start loop FOR from i to 0 till size of an array. Inside the loop, check IF i not equals to temp_2 then check IF i not equals to temp then puch_back arr[i] to vec_1.

• ELSE IF, vec_3 >= 2 then start loop FOR from i to 0 till vec_3 size and check IF abs(arr[vec_3[i]]) greater than N_max then set temp to vec_3[i] and N_max to abs(arr[temp]).

• Start loop FOR from i to 0 till i less tha vec_3.size() and check IF abs(arr[vec_3[i]]) less than N_min than check IF vec_3[i] not equals to temp than set temp_2 to vec_3[i] and N_min to abs(arr[temp_2]).

• Start loop FOR from i to 0 till size of an array. Check IF i not equals to temp_2 then check IF i not equals to temp then push_back arr[i] to vc_1 and keep printing.

## Example

#include <bits/stdc++.h>
using namespace std;
void Minimize_root(int arr[], int size){
vector<int> vec_1;
vector<int> vec_2;
vector<int> vec_3;
for (int i = 0; i < size; i++){
if (arr[i] > 0){
vec_2.push_back(i);
}
else if (arr[i] < 0){
vec_3.push_back(i);
}
}
int vec_2_size = vec_2.size();
int vec_3_size = vec_3.size();

if(vec_2_size >= 2){
if(vec_3_size>= 2){
int max_val = INT_MIN; //max_val
int temp = -1; //temp
int min_val = INT_MAX; //min_val
int temp_2 = -1; //temp_2
int N_max = INT_MIN; // N_max
int N_temp = -1; // N_temp
int N_min = INT_MAX; //N_min
int N_temp_2 = -1; //N_temp_2

for (int i = 0; i < vec_2.size(); i++){
if (arr[vec_2[i]] > max_val){
temp = vec_2[i];
max_val = arr[temp];
}
}

for (int i = 0; i < vec_2.size(); i++){
if (arr[vec_2[i]] < min_val){
if(vec_2[i] != temp){
temp_2 = vec_2[i];
min_val = arr[temp_2];
}
}
}
for (int i = 0; i < vec_3.size(); i++){
if (abs(arr[vec_3[i]]) > N_max){
N_temp = vec_3[i];
N_max = abs(arr[N_temp]);
}
}
for (int i = 0; i < vec_3.size(); i++){
if(abs(arr[vec_3[i]]) < N_min ){
if(vec_3[i] != N_temp){
N_temp_2 = vec_3[i];
N_min = abs(arr[N_temp_2]);
}
}
}

double vec_2_data = -1.0 * (double)max_val / (double)min_val;
double vec_3_data = -1.0 * (double)N_max / (double)N_min;

if (vec_2_data < vec_3_data){
vec_1.push_back(arr[temp_2]);
vec_1.push_back(arr[temp]);
for (int i = 0; i < size; i++){
if (i != temp_2 && i != temp){
vec_1.push_back(arr[i]);
}
}
}
else{
vec_1.push_back(arr[N_temp_2]);
vec_1.push_back(arr[N_temp]);

for (int i = 0; i < size; i++){
if (i != N_temp_2 && i != N_temp){
vec_1.push_back(arr[i]);
}
}
}
for (int i = 0; i < vec_1.size(); i++){
cout << vec_1[i] << " ";
}
}
}
else if(vec_2_size >= 2){
int max_val = INT_MIN;
int temp = -1;
int min_val = INT_MAX;
int temp_2 = -1;
for (int i = 0; i < vec_2.size(); i++){
if (arr[vec_2[i]] > max_val){
temp = vec_2[i];
max_val = arr[temp];
}
}
for (int i = 0; i < vec_2.size(); i++){
if (arr[vec_2[i]] < min_val){
if(vec_2[i] != temp){
temp_2 = vec_2[i];
min_val = arr[temp_2];
}
}
}
vec_1.push_back(arr[temp_2]);
vec_1.push_back(arr[temp]);
int i = 0;
i < size; i++; {
if(i != temp_2){
if(i != temp){
vec_1.push_back(arr[i]);
}
}
}
for (int i = 0; i < vec_1.size(); i++){
cout << vec_1[i] << " ";
}
}
else if(vec_3_size >= 2){
int N_max = INT_MIN;
int temp = -1;
int N_min = INT_MAX;
int temp_2 = -1;
for (int i = 0; i < vec_3.size(); i++){
if (abs(arr[vec_3[i]]) > N_max){
temp = vec_3[i];
N_max = abs(arr[temp]);
}
}
for (int i = 0; i < vec_3.size(); i++){
if(abs(arr[vec_3[i]]) < N_min){
if(vec_3[i] != temp){
temp_2 = vec_3[i];
N_min = abs(arr[temp_2]);
}
}
}
vec_1.push_back(arr[temp_2]);
vec_1.push_back(arr[temp]);
for (int i = 0; i < size; i++){
if (i != temp_2){
if(i != temp){
vec_1.push_back(arr[i]);
}
}
}
for (int i = 0; i < vec_1.size(); i++){
cout << vec_1[i] << " ";
}
} else {
cout<<"Not Possible";
}
}
int main(){
int arr[] = { 2, -1, 4, 9, -1, 10, -5};
int size = sizeof(arr) / sizeof(arr);
cout<<"Minimize the sum of roots of a given polynomial is: ";
Minimize_root(arr, size);
return 0;
}

## Output

If we run the above code it will generate the following Output

Minimize the sum of roots of a given polynomial is: -1 -5 2 4 9 -1 10