Lecture 164:- Polynomial Evaluation

Polynomial evaluation is the process of computing the value of a given polynomial for a specific value of the variable 'x'. 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 whose value we want to evaluate the polynomial for.
• n is the degree of the polynomial, which is the highest power of x in the polynomial.

To evaluate a polynomial for a specific value of 'x', we simply substitute that value into the polynomial expression and perform the arithmetic operations to get the result.

Here's a simple C++ code to evaluate a polynomial for a given value of 'x' using the standard polynomial representation:

cppCopy code

#include <iostream> #include <cmath> using namespace std; // Function to evaluate the polynomial for a given value of x double evaluate_polynomial(const double* coefficients, int degree, double x) {    double result = 0.0;    double power_of_x = 1.0;    for (int i = 0; i <= degree; i++) {        result += coefficients[i] * power_of_x;        power_of_x *= x;    }    return result; } int main() {    // Example: P(x) = 3x^4 + 2x^3 - 5x^2 + 7x + 1    const double coefficients[] = { 1, 7, -5, 2, 3 };    int degree = sizeof(coefficients) / sizeof(coefficients[0]) - 1;    double x = 2.5;    double result = evaluate_polynomial(coefficients, degree, x);    cout << "P(" << x << ") = " << result << endl;    return 0; }

In this code, the evaluate_polynomial function takes an array of coefficients, the degree of the polynomial, and the value of 'x' as inputs. It then computes the result of the polynomial for the given value of 'x' and returns the result. The main function evaluates the polynomial P(x) = 3x^4 + 2x^3 - 5x^2 + 7x + 1 for x = 2.5 and prints the result.