Synthetic Division

Synthetic division is a simplified method used to divide polynomials, particularly when dividing by linear factors. It involves a step-by-step process where coefficients of the polynomial are manipulated without explicitly writing down variables. This article provides an in-depth explanation of synthetic division further it covers the method’s definition, comparison with long division, step-by-step procedures, examples, advantages, disadvantages, and practice questions.

What is Synthetic Division?

Synthetic division is a shortcut method for dividing a polynomial by a term (x – c), where c is a specific number. It’s an easier and faster way to divide polynomials, especially when dividing by linear factors, instead of using the usual long division method.

Synthetic Division of Polynomials Definition

Synthetic Division is an algebraic concept in Mathematics used to divide polynomials especially when the divisor is in the form of (x – c).

It is generally used to find the zeros of a polynomial. It is considered better as it is a faster, simpler and more effective way of solving some division problems when compared to the traditional methods for the same.

Synthetic Division Method

Synthetic Division is a systematic approach to solve a problem. For solving a question you need to identify the divisor and consider the coefficients of the polynomial. Perform synthetic division and then derive the solution for the same. This method involves a set of basic arithmetic operations and this makes it an easy way to solve the problem when compared to the traditional method for division.

How to do Synthetic Division?

Below will be step-by-step solution for the division of any linear polynomial with the help of synthetic division method.

Let us consider the previous example only and solve it step by step i.e. dividing 4x² – 6x -8 by x – 2.

Step 1: Identifying Co-Efficient.

Write down all the coefficients including the missing term. If some term is missing write 0 in its place.

4 -6 -8

Step 2: Identify the divisor.

Here, (x-2) is divisor so x -2 =0 ⇒ x = +2. Therefore, 2 is the divisor.

2 | 4 -6 -8

Step 3: Write first co-efficient as it is.

Write down the first coefficient as it is here 4 is the first co-efficient.

Step 4: Multiply and add it with divisor.

Multiply the first coefficient with the divisor here 2*4 and write the result below second co-efficient and add it and write down the result.

Step 5: Repeat the step 4.

Step 6: Derive Result

Result will depend on the last line of the solution given above. Here, the result will be

4x + 2 – 4/(x-2)

Synthetic Division Vs Long Division

The key difference between synthetic division and long division are:

Aspect	Synthetic Division	Long Division
Purpose	Quick division of polynomials by a linear divisor	Division of polynomials by any divisor
Requirement	Divisor must be linear (degree 1)	Divisor can be of any degree
Division process	Uses coefficients of the polynomial	Involves dividing each term of the polynomial
Algorithm efficiency	Faster, especially for linear divisors	Slower compared to synthetic division
Steps involved	Fewer steps	More steps, including bringing down terms
Usage	Suitable for linear divisors in polynomial long division	Suitable for any kind of polynomial division

Advantages and Disadvantages of Synthetic Division Method

Synthetic division is a specialized technique used to divide polynomials efficiently. While it offers speed and simplicity for certain cases, its applicability is limited to specific polynomial divisions.

Advantages	Disadvantages
Synthetic division is often faster and more straightforward than polynomial long division	Synthetic division is only applicable when dividing polynomials by linear factors
Synthetic division involves a relatively simple process of writing down coefficients and performing basic arithmetic operations	Synthetic division is a specialized technique tailored for specific types of polynomial division problems
Synthetic division is particularly useful when dividing polynomials by linear factors of the form (x – c)	To use synthetic division effectively, one must have a good understanding of polynomial coefficients, linear factors, and the process involved
It is commonly used in solving polynomial equations, analyzing polynomial functions, and simplifying algebraic expressions.	Synthetic division is generally less prone to errors than traditional long division, mistakes can still occur, especially if coefficients are written incorrectly

Solved Examples on Synthetic Division

Example 1: Divide x³ – 5x² + 6x + 7 by x – 3.

Solution:

Thus, x³ – 5x² + 6x + 7 by x – 3 = x² – 2x + 7/(x-3)

Example 2: Divide 5x² + 3x – 2 by x + 2.

Solution:

Thus, 5x² + 3x – 2 by x + 2 = 5x² – 7x + 12/(x+2).

Example 3: Divide x³ – 5x – 9 by x – 4.

Solution:

Thus, x³ – 5x – 9 by x – 4 = x² + 4x + 11 + 35/(x – 4).

Practice Questions on Synthetic Division

Q1: Divide x³+ 4x² -8x +5 by x – 3.

Q2: Divide 7x³ -9x + 4 by x + 4.

Q3: Divide -4x³ + 2x² +6x by x – 2.

Q4: Divide -x³ -3x +8 by x – 6.

Q5: Divide 4x² -x +8 by x + 5.

Synthetic Division: FAQs

What is synthetic division?

Synthetic division is a method used to divide a polynomial by a linear factor of the form x − c, where c is a constant.

When should we use synthetic division?

Synthetic division is particularly useful when dividing polynomials by linear factors, especially when the divisor is of the form x − c.

Can synthetic division be used for all polynomial divisions?

No, synthetic division is specifically designed for dividing polynomials by linear factors.

Are there any limitations of synthetic division?

Synthetic division can only be used for dividing polynomials by linear factors of the form x − c.