Lecture 165:- Polynomial Addition

Polynomial addition involves adding two polynomials together to create a new polynomial. The process is straightforward and involves combining like terms with the same degree. When performing polynomial addition, you add the coefficients of the terms with the same degree and keep all other terms intact.

Here's the general form of two polynomials that we want to add:

P(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_2 * x^2 + a_1 * x + a_0 Q(x) = b_m * x^m + b_(m-1) * x^(m-1) + ... + b_2 * x^2 + b_1 * x + b_0

The sum polynomial, R(x), will be:

R(x) = (a_n + b_m) * x^max(n, m) + (a_(n-1) + b_(m-1)) * x^(max(n, m) - 1) + ... + (a_2 + b_2) * x^2 + (a_1 + b_1) * x + (a_0 + b_0)

To perform polynomial addition, you need to compare the degrees of the two polynomials and determine the maximum degree (max(n, m)). Then, you add the coefficients of the corresponding terms for each degree from the highest degree to the lowest degree.

Here's a C++ code to perform polynomial addition:

cppCopy code

#include <iostream> #include <vector> using namespace std; // Structure to represent a term in the polynomial struct Term {    int degree;    int coefficient; }; // Function to perform polynomial addition vector<Term> add_polynomials(const vector<Term>& polynomial1, const vector<Term>& polynomial2) {    vector<Term> result;    int i = 0;    int j = 0;    // Iterate through both polynomials until one of them is fully processed    while (i < polynomial1.size() && j < polynomial2.size()) {        if (polynomial1[i].degree == polynomial2[j].degree) {            // Coefficients are added if the degrees are the same            int sum_coeff = polynomial1[i].coefficient + polynomial2[j].coefficient;            if (sum_coeff != 0) {                result.push_back({polynomial1[i].degree, sum_coeff});            }            i++;            j++;        } else if (polynomial1[i].degree > polynomial2[j].degree) {            result.push_back(polynomial1[i]);            i++;        } else {            result.push_back(polynomial2[j]);            j++;        }    }    // Copy any remaining terms from polynomial1    while (i < polynomial1.size()) {        result.push_back(polynomial1[i]);        i++;    }    // Copy any remaining terms from polynomial2    while (j < polynomial2.size()) {        result.push_back(polynomial2[j]);        j++;    }    return result; } // Function to print the polynomial void print_polynomial(const vector<Term>& polynomial) {    for (int i = 0; i < polynomial.size(); i++) {        cout << polynomial[i].coefficient << "x^" << polynomial[i].degree;        if (i < polynomial.size() - 1) {            cout << " + ";        }    }    cout << endl; } int main() {    // Example polynomials:    // P(x) = 3x^4 + 2x^3 - 5x^2 + 7x + 1    vector<Term> polynomial1 = { {4, 3}, {3, 2}, {2, -5}, {1, 7}, {0, 1} };        // Q(x) = 2x^3 - 3x^2 + 4x + 5    vector<Term> polynomial2 = { {3, 2}, {2, -3}, {1, 4}, {0, 5} };    cout << "Polynomial P(x): ";    print_polynomial(polynomial1);    cout << "Polynomial Q(x): ";    print_polynomial(polynomial2);    vector<Term> result = add_polynomials(polynomial1, polynomial2);    cout << "Result Polynomial R(x): ";    print_polynomial(result);    return 0; }

In this C++ code, the Term structure represents a term in the polynomial with a degree and a coefficient. The add_polynomials function takes two polynomials as input and returns their sum as a new polynomial. The print_polynomial function is used to print a polynomial in a formatted manner.

The main function demonstrates the usage of the add_polynomials function to add two example polynomials P(x) and Q(x) and prints the result polynomial R(x).