C Program for Synthetic Division with algorithm and example

Algorithm of Synthetic Division:

Given a polynomial of form p(x) = a_nxⁿ + a_n-1x^n-1+…+ a₁x+ a₀, we can divide it by a linear factor x-r, where ‘r’ is a constant, using following steps.

a.       Write the r on the left side of a vertical bar and the coefficients in decreasing degree.

r| a_n               a_n-1         …….        a₁            a₀

b.      Pass the coefficient of the highest degree below the horizontal line as it is

 r| a_n              a_n-1         …….        a₁            a₀

  |.                                                                   .

  | a_n             

c.       Multiply the passed coefficient with ‘r’ and add to the next  coefficient

 r| a_n              a_n-1         …….        a₁            a₀

  |.                 a_n                                                  .

  | a_n          a_n-1+ a_n

d.      Continue the step ‘c’ until all coefficients are covered.

e.      The numbers below a_n ……. a₁ give the coefficients of the quotient whose degree is 1 less than the polynomial, the number below a₀ is the remainder.

Example of Synthetic Division:  x² + 5x+ 6 by x – 1

Writing the coefficients of the polynomial and dividing by the linear factor.

1|1         5              6

  |           1              6

  |1         6              12

Therefore, the quotient Q(x) = x + 6 and remainder R(x) = 12.

Program:

#include<stdio.h>

#include<conio.h>

void main(){

                clrscr();

                int poly[6], m, r, i, q[6];

                printf("\t\tSYNTHETIC DIVISION");

                printf("\n Enter the highest degree of the equation (max 5): ");

                scanf("%d",&m);

                for(i=0;i<=m;i++){

                                printf("\n Coefficient x[%d] = ", m-i);

                                scanf("%d",&poly[i]);

                }

                printf("\n Enter the value of constant in (x-r) : ");

                scanf("%d",&r);

                q[0] = poly[0];

                for(i=1;i<=m;i++){

                                q[i] = (q[i-1]*r)+poly[i];

                }

                printf("\n Coefficients of quotient are: \n");

                for(i=0;i<m;i++){

                                printf("\t%d",q[i]);

                }

                printf("\n Remainder is: %d", q[m]);

getch();

}

Sample Output:

SYNTHETIC DIVISION
Enter the highest degree of the equation (max 5): 2

Coefficient x[0] = 1

Coefficient x[1] = 5

Coefficient x[2] = 6

Enter the value of constant in (x—r) : 1

Coefficients of quotient are:
1        6
Remainder is: 12

©Dixit Bhatta 2013