Vieta’s formula relates the coefficients of polynomial to the sum and product of their roots, as well as the products of the roots taken in groups. Vieta’s formula describes the relationship of the roots of a polynomial with its coefficients. Consider the following example to find a polynomial with given roots. (Only discuss real-valued polynomials, i.e. the coefficients of polynomials are real numbers). Let’s take a quadratic polynomial. Given two real roots and
, then find a polynomial.
Consider the polynomial . Given the roots, we can also write it as
.
Since both equation represents the same polynomial, so equate both polynomial
Simplifying the above equation, we get
Comparing the coefficients of both sides, we get
For ,
,
For ,
,
For constant term, ,
Which gives,
,
Equation (1) and (2) are known as Vieta’s Formulas for a second degree polynomial.
In general, for an degree polynomial, there are n different Vieta’s Formulas. They can be written in a condensed form as
For
The following examples illustrate the use of Vieta’s formula to solve a problem.
Examples:
Input : n = 2 roots = {-3, 2} Output : Polynomial coefficients: 1, 1, -5 Input : n = 4 roots = {-1, 2, -3, 7} Output : Polynomial coefficients: 1, -5, -19, 29, 42
C++
// C++ program to implement vieta formula // to calculate polynomial coefficients. #include <bits/stdc++.h> using namespace std; // Function to calculate polynomial // coefficients. void vietaFormula( int roots[], int n) { // Declare an array for // polynomial coefficient. int coeff[n + 1]; // Set all coefficients as zero initially memset (coeff, 0, sizeof (coeff)); // Set highest order coefficient as 1 coeff[n] = 1; for ( int i = 1; i <= n; i++) { for ( int j = n - i - 1; j < n; j++) { coeff[j] = coeff[j] + (-1) * roots[i - 1] * coeff[j + 1]; } } cout << "Polynomial Coefficients: " ; for ( int i = n; i >= 0; i--) { cout << coeff[i] << " " ; } } // Driver code int main() { // Degree of required polynomial int n = 4; // Initialise an array by // root of polynomial int roots[] = { -1, 2, -3, 7 }; // Function call vietaFormula(roots, n); return 0; } |
Java
// Java program to implement vieta formula // to calculate polynomial coefficients. import java.util.Arrays; class GFG { // Function to calculate polynomial // coefficients. static void vietaFormula( int roots[], int n) { // Declare an array for // polynomial coefficient. int coeff[] = new int [++n + 1 ]; Arrays.fill(coeff, 0 ); // Set highest order coefficient as 1 coeff[n] = 1 ; for ( int i = 1 ; i <n; i++) { for ( int j = n - i - 1 ; j < n; j++) { coeff[j] = coeff[j] + (- 1 ) * roots[i - 1 ] * coeff[j + 1 ]; } } System.out.print( "Polynomial Coefficients: " ); for ( int i = n; i > 0 ; i--) { System.out.print(coeff[i] + " " ); } } // Driver code public static void main(String[] args) { // Degree of required polynomial int n = 4 ; // Initialise an array by // root of polynomial int roots[] = { - 1 , 2 , - 3 , 7 }; // Function call vietaFormula(roots, n); } } /* This code contributed by PrinciRaj1992 */ |
Python3
# Python3 program to implement # Vieta's formula to calculate # polynomial coefficients. def vietaFormula(roots, n): # Declare an array for # polynomial coefficient. coeff = [ 0 ] * (n + 1 ) # Set Highest Order # Coefficient as 1 coeff[n] = 1 for i in range ( 1 , n + 1 ): for j in range (n - i - 1 , n): coeff[j] + = (( - 1 ) * roots[i - 1 ] * coeff[j + 1 ]) # Reverse Array coeff = coeff[:: - 1 ] print ( "Polynomial Coefficients : " , end = "") # Print Coefficients for i in coeff: print (i, end = " " ) print () # Driver Code if __name__ = = "__main__" : # Degree of Polynomial n = 4 # Initialise an array by # root of polynomial roots = [ - 1 , 2 , - 3 , 7 ] # Function call vietaFormula(roots, n) # This code is contributed # by Arihant Joshi |
C#
// C# program to implement vieta formula // to calculate polynomial coefficients. using System; class GFG { // Function to calculate polynomial // coefficients. static void vietaFormula( int []roots, int n) { // Declare an array for // polynomial coefficient. int []coeff = new int [++n + 1]; // Set highest order coefficient as 1 coeff[n] = 1; for ( int i = 1; i <n; i++) { for ( int j = n - i - 1; j < n; j++) { coeff[j] = coeff[j] + (-1) * roots[i - 1] * coeff[j + 1]; } } Console.Write( "Polynomial Coefficients: " ); for ( int i = n; i > 0; i--) { Console.Write(coeff[i] + " " ); } } // Driver code public static void Main(String[] args) { // Degree of required polynomial int n = 4; // Initialise an array by // root of polynomial int []roots = { -1, 2, -3, 7 }; // Function call vietaFormula(roots, n); } } // This code has been contributed by 29AjayKumar |
PHP
<?php // PHP program to implement vieta formula // to calculate polynomial coefficients. // Function to calculate polynomial // coefficients. function vietaFormula( $roots , $n ) { // Declare an array for // polynomial coefficient. $coeff = array_fill (0, $n + 1, 0); // Set all coefficients as zero initially // Set highest order coefficient as 1 $coeff [ $n ] = 1; for ( $i = 1; $i <= $n ; $i ++) { for ( $j = $n - $i ; $j < $n ; $j ++) { $coeff [ $j ] = $coeff [ $j ] + (-1) * $roots [ $i - 1] * $coeff [ $j + 1]; } } echo "polynomial coefficients: " ; for ( $i = $n ; $i >= 0; $i --) { echo $coeff [ $i ]. " " ; } } // Driver code // Degree of required polynomial $n = 4; // Initialise an array by // root of polynomial $roots = array (-1, 2, -3, 7); // Function call vietaFormula( $roots , $n ); // This code is contributed by mits ?> |
Polynomial Coefficients: 1 -5 -19 29 42
Time Complexity : .
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