Lecture 163:- Polynomial Representation

Polynomial representation is a mathematical method used to express a polynomial function. A polynomial is an expression consisting of variables (usually represented by the letter 'x') raised to non-negative integer powers, combined with coefficients. The general form of a polynomial is:

P(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_2 * x^2 + a_1 * x + a_0

Where:

• P(x) is the polynomial function.
• a_n, a_(n-1), ..., a_2, a_1, and a_0 are the coefficients of the polynomial.
• x is the variable.
• n is the degree of the polynomial, which is the highest power of x in the polynomial.

Polynomials can be represented in different forms, and two common representations are:

1. Standard Polynomial Representation: In the standard representation, we write down the polynomial in its expanded form, showing all the terms explicitly. For example, the polynomial P(x) = 3x^4 + 2x^3 - 5x^2 + 7x + 1 is represented in standard form.
2. Coefficient List Representation: In this representation, we store the coefficients of the polynomial in an array or list. Each index of the array corresponds to the power of x in the polynomial. The coefficient at index i represents the coefficient of x^i. For example, the polynomial P(x) = 3x^4 + 2x^3 - 5x^2 + 7x + 1 can be represented as [3, 2, -5, 7, 1].

The coefficient list representation is particularly useful when dealing with large polynomials with many zero coefficients, as it can save memory and computation time.

Here's a simple C++ code to represent a polynomial using the coefficient list representation:

cppCopy code

#include <iostream> #include <vector> using namespace std; // Function to evaluate the polynomial for a given value of x int evaluate_polynomial(const vector<int>& coefficients, int x) {    int result = 0;    int power_of_x = 1;    for (int i = 0; i < coefficients.size(); i++) {        result += coefficients[i] * power_of_x;        power_of_x *= x;    }    return result; } int main() {    vector<int> coefficients = {3, 2, -5, 7, 1};    int x = 2;    int result = evaluate_polynomial(coefficients, x);    cout << "P(" << x << ") = " << result << endl;    return 0; }

In this code, the evaluate_polynomial function takes the coefficients of the polynomial and the value of x as inputs and calculates the result of the polynomial for the given value of x. The main function evaluates the polynomial P(x) = 3x^4 + 2x^3 - 5x^2 + 7x + 1 for x = 2 and prints the result.