A polynomial is a mathematical expression consisting of multiple terms, with each term comprising a coefficient and an exponent. The standard representation of a polynomial is:
P(x)=a n x n +a n-1 x n-1 +…+a 1 x+a 0
X represents the variable in this context, while the coefficients are denoted as a n, a n-1, ..., a 0. The polynomial is of degree n, with the leading term being the one that contains the highest exponent of n within the polynomial.
Example:
Let's consider an example to grasp the concept of adding polynomials in the C programming language:
#include <stdio.h>
#include <stdlib.h>
struct Term
{
int coefficient;
int exponent;
};
struct Polynomial
{
int numTerms;
struct Term *terms;
};
void createPolynomial(struct Polynomial *poly)
{
int i;
printf("Enter the number of terms: ");
scanf("%d", &poly->numTerms);
poly->terms = (struct Term *)malloc(poly->numTerms * sizeof(struct Term));
printf("Enter the terms:\n");
for (i = 0; i< poly->numTerms; i++)
scanf("%d%d", &poly->terms[i].coefficient, &poly->terms[i].exponent);
printf("\n");
}
void displayPolynomial(struct Polynomial poly)
{
int i;
for (i = 0; i<poly.numTerms; i++)
{
printf("%dx^%d", poly.terms[i].coefficient, poly.terms[i].exponent);
if (i + 1 <poly.numTerms)
printf(" + ");
}
printf("\n");
}
struct Polynomial *addPolynomials(struct Polynomial *poly1, struct Polynomial *poly2)
{
int i, j, k;
struct Polynomial *sum;
sum = (struct Polynomial *)malloc(sizeof(struct Polynomial));
sum->terms = (struct Term *)malloc((poly1->numTerms + poly2->numTerms) * sizeof(struct Term));
i = j = k = 0;
while (i< poly1->numTerms&& j < poly2->numTerms)
{
if (poly1->terms[i].exponent> poly2->terms[j].exponent)
sum->terms[k++] = poly1->terms[i++];
else if (poly1->terms[i].exponent< poly2->terms[j].exponent)
sum->terms[k++] = poly2->terms[j++];
else
{
sum->terms[k].exponent = poly1->terms[i].exponent;
sum->terms[k++].coefficient = poly1->terms[i++].coefficient + poly2->terms[j++].coefficient;
}
}
for (; i< poly1->numTerms; i++)
sum->terms[k++] = poly1->terms[i];
for (; j < poly2->numTerms; j++)
sum->terms[k++] = poly2->terms[j];
sum->numTerms = k;
return sum;
}
int main()
{
struct Polynomial poly1, poly2, *polySum;
printf("Enter Polynomial 1:\n");
createPolynomial(&poly1);
printf("Enter Polynomial 2:\n");
createPolynomial(&poly2);
polySum = addPolynomials(&poly1, &poly2);
printf("\n");
printf("Polynomial 1 is: ");
displayPolynomial(poly1);
printf("\n");
printf("Polynomial 2 is: ");
displayPolynomial(poly2);
printf("\n");
printf("Sum of the polynomials is: ");
displayPolynomial(*polySum);
free(poly1.terms);
free(poly2.terms);
free(polySum->terms);
free(polySum);
return 0;
}
Output:
Enter Polynomial 1:
Enter the number of terms: 3
Enter the terms:
3 2
3 1
1 3
Enter Polynomial 2:
Enter the number of terms: 3
Enter the terms:
1 2
2 1
3 2
Polynomial 1 is: 3x^2 + 3x^1 + 1x^3
Polynomial 2 is: 1x^2 + 2x^1 + 3x^2
Sum of the polynomials is: 4x^2 + 5x^1 + 1x^3 + 3x^2
Explanation:
This C program is dedicated to performing polynomial addition through a custom data structure. It is organized into functions that facilitate the generation, presentation, and merging of two polynomials. The struct Term and struct Polynomial are employed to encapsulate the elements and attributes of a polynomial. The createPolynomial function guides the user to enter the quantity of terms, coefficients, and exponents of each term for both polynomials. The information is dynamically assigned to the terms array inside each polynomial structure.
The displayPolynomial method loops through the array of terms in a specified polynomial, displaying the coefficient and exponent of each term. The primary functionality of the software is centered around the addPolynomials method, which accepts two polynomial structures and adds their terms together. Within the main function, users are prompted to enter coefficients and exponents for two polynomials. Subsequently, the addPolynomials method calculates the total of the two polynomials, storing the result in the polySum structure. The outcomes are exhibited through the displayPolynomial routine.
The software demonstrates the process of dynamic memory allocation to construct and modify polynomial structures, aiding in the comprehension of data structure manipulation and mathematical operations. Memory allocated while the program runs is released appropriately with the free function to avoid memory leaks.