Theory:

Synthetic division is a shortcut method of polynomial division used to find the roots of the polynomial.

Let us learn the working rule to perform synthetic division.
Working rule:

Step 1: Arrange the terms of the polynomial $$p(x)$$ (Dividend) in the standard form.

Step 2: Write the coefficients of the variables along with the constant of the dividend in the first row and put $$0$$ for the missing terms.

Step 3: Find the zero of the divisor $$d(x)$$.

Step 4: Write the zero of the divisor in front of the dividend in the first row and put $$0$$ in the second row of first column.

Step 5: Multiply the sum obtained from first column with the zero of the divisor to fill up the second row of second column and continue the same process to complete the second and third row.
Let us understand the process of synthetic division using an example.
Example:
Let the polynomial $$p(x)$$ $$=$$ $$x^3 - 7x + x^2 - 3$$ be the dividend and $$d(x)$$ $$=$$ $$-3 + x$$ be the divisor.

We will find the quotient and the remainder using synthetic division as follows:

Step 1: Arrange the terms of the polynomial in the standard form as follows:

$$p(x)$$ $$=$$ $$x^3 + x^2 - 7x - 3$$

$$d(x)$$ $$=$$ $$x - 3$$

Step 2: Write the coefficients of the variables along with the constant of the dividend in the first row and put $$0$$ for the missing terms as follows:

$$\begin{array}{r|rrrrr} & 1 & 1 & -7 & -3 \\& & & & \\\hline & & & & \end{array}$$

Step 3: Find the zero of the divisor $$d(x)$$ $$=$$ $$x - 3$$ as follows:

Equate the divisor to zero and solve for $$x$$.

$$x - 3 = 0$$

$$x$$ $$=$$ $$3$$

Step 4: Write the zero of the divisor $$3$$ in front of the dividend in the first row and put $$0$$ in the second row of first column as shown below:

$$\begin{array}{r|rrrrr}{3} & 1 & 1 & -7 & -3 \\& 0 & & & \\\hline & & & & \end{array}$$

Step 5: Multiply the sum obtained from first column with the zero of the divisor to fill up the second row of second column and continue the same process to complete the second and third row as follows:

$$\begin{array}{r|rrrrr}{3} & 1 & 1 & -7 & -3 \\& 0 & 3 & 12 & 15 \\\hline & 1 & 4 & 5 &\underline{\begin{array}{|r} {12} \end {array}} \end{array}$$

The quotient and the remainder is obtained from the third row.

Here, the quotient is $$x^2+4x+5$$ and the remainder is $$12$$.