Explain the Addition of Two variable polynomial

Polynomials are fundamental in mathematics, and their use extends into computer science, engineering, physics, and more. While single-variable polynomials are common, two-variable polynomials provide greater flexibility and represent complex relationships in multidimensional space.

What is a Two-Variable Polynomial?

A two-variable polynomial is a mathematical expression involving two variables, typically x and y, with non-negative integer exponents.

Example:

P(x, y) = 3x²y + 4xy² + 5
Q(x, y) = 2x²y + 6xy² + 7

When adding P(x, y) and Q(x, y), we combine like terms (same powers of x and y):

Result = (3+2)x²y + (4+6)xy² + (5+7)
       = 5x²y + 10xy² + 12

Understanding Like Terms

• Like terms have identical exponents for both x and y.
• Only like terms can be combined by summing their coefficients.

For instance:

• 3x²y and 2x²y are like terms.
• 3xy² and 4x²y are not like terms.

How to Represent a Polynomial in C?

To store and manipulate polynomials programmatically, we define each term using a structure that includes:

• coefficient
• exponent of x (expX)
• exponent of y (expY)

struct Term {
    int coeff;
    int expX;
    int expY;
};

The entire polynomial is then represented as an array of Term.

#include <stdio.h>
#include <stdlib.h>

// Define structure for each term in the polynomial
typedef struct {
    int coeff;
    int expX;
    int expY;
} Term;

// Function to input a polynomial
void inputPolynomial(Term poly[], int n) {
    for (int i = 0; i < n; i++) {
        printf("Enter coefficient, exponent of x, and exponent of y for term %d: ", i + 1);
        scanf("%d %d %d", &poly[i].coeff, &poly[i].expX, &poly[i].expY);
    }
}

// Function to add two polynomials
int addPolynomials(Term p1[], int n1, Term p2[], int n2, Term result[]) {
    int i, j, k = 0;
    int used[n2];
    for (i = 0; i < n2; i++) used[i] = 0;

    // Combine like terms
    for (i = 0; i < n1; i++) {
        int found = 0;
        for (j = 0; j < n2; j++) {
            if (p1[i].expX == p2[j].expX && p1[i].expY == p2[j].expY) {
                result[k].coeff = p1[i].coeff + p2[j].coeff;
                result[k].expX = p1[i].expX;
                result[k].expY = p1[i].expY;
                used[j] = 1;
                found = 1;
                k++;
                break;
            }
        }
        if (!found) {
            result[k++] = p1[i];
        }
    }

    // Add remaining terms from second polynomial
    for (j = 0; j < n2; j++) {
        if (!used[j]) {
            result[k++] = p2[j];
        }
    }

    return k; // number of terms in the result
}

// Function to display a polynomial
void displayPolynomial(Term poly[], int n) {
    for (int i = 0; i < n; i++) {
        printf("%+d*x^%d*y^%d ", poly[i].coeff, poly[i].expX, poly[i].expY);
    }
    printf("\n");
}

// Main function
int main() {
    int n1, n2;
    printf("Enter number of terms in first polynomial: ");
    scanf("%d", &n1);
    Term poly1[n1];
    inputPolynomial(poly1, n1);

    printf("Enter number of terms in second polynomial: ");
    scanf("%d", &n2);
    Term poly2[n2];
    inputPolynomial(poly2, n2);

    Term result[n1 + n2]; // max possible size
    int resSize = addPolynomials(poly1, n1, poly2, n2, result);

    printf("\nFirst Polynomial:\n");
    displayPolynomial(poly1, n1);

    printf("Second Polynomial:\n");
    displayPolynomial(poly2, n2);

    printf("Sum of Polynomials:\n");
    displayPolynomial(result, resSize);

    return 0;
}

Output

Enter number of terms in first polynomial: 2
Enter coefficient, exponent of x, and exponent of y for term 1: 1 5 7
Enter coefficient, exponent of x, and exponent of y for term 2: 9 4 1
Enter number of terms in second polynomial: 2
Enter coefficient, exponent of x, and exponent of y for term 1: 9 3 1
Enter coefficient, exponent of x, and exponent of y for term 2: 8 5 1

First Polynomial:
+1*x^5*y^7 +9*x^4*y^1 
Second Polynomial:
+9*x^3*y^1 +8*x^5*y^1 
Sum of Polynomials:
+1*x^5*y^7 +9*x^4*y^1 +9*x^3*y^1 +8*x^5*y^1