Multiplications of Two variable polynomial

Polynomial multiplication is a common operation in algebra, used in areas such as scientific computing, engineering simulations, and symbolic algebra systems. When working with two-variable polynomials, the process involves combining like terms and multiplying each term of one polynomial with every term of another.

What Is a Two-Variable Polynomial?

A two-variable polynomial takes the form:

P(x, y) = a₁x^i y^j + a₂x^k y^l + ...

Each term consists of:

• A coefficient
• An exponent for variable x
• An exponent for variable y

How Does Polynomial Multiplication Work?

Let’s take two example polynomials:

P1(x, y) = 2x + 3y  
P2(x, y) = x + y

Multiplication (P1 × P2):

Multiply each term of P1 by each term of P2:

= (2x)(x) + (2x)(y) + (3y)(x) + (3y)(y)
= 2x² + 2xy + 3xy + 3y²
= 2x² + 5xy + 3y²

The final result combines like terms (in this case, 2xy + 3xy = 5xy).

C Structure to Represent Terms

We’ll define a Term structure to hold each term’s data:

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

We’ll use arrays of terms to represent each polynomial.

C Program to Multiply Two-Variable Polynomials

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

#define MAX 100

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

// Input a polynomial
int inputPolynomial(Term poly[]) {
    int n;
    printf("Enter number of terms: ");
    scanf("%d", &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);
    }
    return n;
}

// Display polynomial
void displayPolynomial(Term poly[], int n) {
    for (int i = 0; i < n; i++) {
        if (poly[i].coeff != 0) {
            printf("%d*x^%d*y^%d", poly[i].coeff, poly[i].expX, poly[i].expY);
            if (i < n - 1)
                printf(" + ");
        }
    }
    printf("\n");
}

// Combine like terms
int combineTerms(Term result[], int n) {
    for (int i = 0; i < n; i++) {
        if (result[i].coeff == 0)
            continue;
        for (int j = i + 1; j < n; j++) {
            if (result[i].expX == result[j].expX && result[i].expY == result[j].expY) {
                result[i].coeff += result[j].coeff;
                result[j].coeff = 0; // Mark as combined
            }
        }
    }

    // Remove terms with 0 coefficient
    Term temp[MAX];
    int k = 0;
    for (int i = 0; i < n; i++) {
        if (result[i].coeff != 0) {
            temp[k++] = result[i];
        }
    }

    for (int i = 0; i < k; i++) {
        result[i] = temp[i];
    }

    return k;
}

// Multiply two polynomials
int multiplyPolynomials(Term p1[], int n1, Term p2[], int n2, Term result[]) {
    int k = 0;
    for (int i = 0; i < n1; i++) {
        for (int j = 0; j < n2; j++) {
            result[k].coeff = p1[i].coeff * p2[j].coeff;
            result[k].expX = p1[i].expX + p2[j].expX;
            result[k].expY = p1[i].expY + p2[j].expY;
            k++;
        }
    }
    return combineTerms(result, k);
}

// Main function
int main() {
    Term poly1[MAX], poly2[MAX], result[MAX];
    int n1, n2, nResult;

    printf("Enter first polynomial\n");
    n1 = inputPolynomial(poly1);

    printf("Enter second polynomial\n");
    n2 = inputPolynomial(poly2);

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

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

    nResult = multiplyPolynomials(poly1, n1, poly2, n2, result);

    printf("\nResult (P1 * P2): ");
    displayPolynomial(result, nResult);

    return 0;
}

Output

Enter first polynomial
Enter number of terms: 2
Enter coefficient, exponent of x and exponent of y for term 1: 4 2 1
Enter coefficient, exponent of x and exponent of y for term 2: 5 3 1
Enter second polynomial
Enter number of terms: 2
Enter coefficient, exponent of x and exponent of y for term 1: 6 3 1
Enter coefficient, exponent of x and exponent of y for term 2: 4 2 1

First Polynomial: 4*x^2*y^1 + 5*x^3*y^1
Second Polynomial: 6*x^3*y^1 + 4*x^2*y^1

Result (P1 * P2): 44*x^5*y^2 + 16*x^4*y^2 + 30*x^6*y^2