Introduction: Synthetic division is one of the ways to manually perform Euclidean division of polynomials in algebra. Polynomial division can also be done using the long division approach. Synthetic division, on the other hand, takes less writing and computations than the long division technique of polynomials. For the exceptional instances of dividing by a linear factor, the synthetic division is the shorter approach to the standard long division of a polynomial. a technique for dividing a polynomial by another polynomial of the first degree that involves merely writing down the coefficients of the various powers of the variable and altering the sign of the constant term in the divisor, thereby replacing the conventional subtractions with additions.

Define Synthetic Division

When the divisor is the linear factor, synthetic division can be evaluated to execute the division operation on polynomials. One of the advantages of utilizing this approach over the standard long method is that it allows you to compute without having to write variables when conducting polynomial division, which makes it a simpler way than long division.

The division of two polynomials may be written as p(x)/q(x) = Q + R/(q(x)).

where p(x) denotes a dividend, The linear divisor is q(x), The letter Q stands for quotient, R stands for “remainder”.

Synthetic division is faster than long division because it requires fewer computations. When computing the steps required in the polynomial division, it takes up significantly less space. Only when the divisor is a linear polynomial may synthetic division be utilized. For the other situations, we must use the long division approach. Some common steps can be used to perform synthetic division. Take the coefficients by themselves, bring the first down, multiply with the linear factor’s zero, then add with the next coefficient, and so on until the finish.

Definition of Synthetic Division of the Polynomials 

On dividing the polynomial p(x) by the linear factor (x – a), and Q(x) be the quotient polynomial and r is the remainder.

p(x)/q(x) = p(x)/ (x- a) = Quotient + (Remainder/ (x – a))

p(x)/ (x – a) = Q(x) + (r/ (x – a))

Example: Let us take an example to understand Synthetic division, Balram is an apple vendor. His gains before the day were x, and today’s profits are ((x * x) – 2). What was the profit per apple if the number of apples he sold was (x + 2)? 

The answer is obtained by rewriting the problem as (x² + x – 2) (x + 2).

• Step 1: Inside the box, write the dividend coefficients and x + 2 as the divisor.
• Step 2: Remove the leading coefficient 1 from the top row and place it in the bottom row.
• Step 3: Multiply -2 by 1 and record the result in the centre row as the product -2.
• Step 4: In the second column, add 1 and -2 and write the sum -1 in the bottom row.
• Step 5: Multiply -2 by -1 (the answer from step 4) and write the product 2 below -2.
• Step 6: In the third column, add -2 and 2 and enter the sum 0 in the bottom row.
• Step 7: The coefficient of the quotient is shown in the bottom row. The quotient has a degree of one less than the dividend. As a result, x – 1 + 0/ (x + 2) = x – 1 is the final solution.

Method of solving Synthetic division of polynomials 

The Synthetic division of the polynomials is the process of eliminating the need for variables by calculating with integers. We multiply instead of division, and we add in place of subtraction

• Firstly, Write the dividend coefficients and replace the divisor with the linear factor’s zero.
• Reduce the first coefficient to its lowest value and multiply it by the divisor.
• Add the column and write the product below the second coefficient.
• Continue until you’ve reached the last coefficient. The remainder is calculated by the last number.
• Write the quotient using the coefficients.
• The dividend polynomial is one order lower than the outcome polynomial.

Point to Remember;

• To get the quotient and the remainder, write down the coefficients and divide them by the linear factor’s zero. Q(x) + (R/(x – a)) = P(x)/(x – a)
• The quotient should be (Q(x)/b) when we apply synthetic division by (bx + a).
• When the divisor is a linear factor, only use synthetic division.
• Multiplication and addition should be employed instead of division and subtraction in the long division approach.
• “Synthetic division is a simple method of dividing a polynomial by another polynomial equation of degree 1 that is commonly used to locate polynomial zeros.”