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.
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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 |
Related Articles,
- Dividing Polynomials & Long Division Algorithm
- Division Algorithm for Polynomials
- Multiplying Polynomials
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.