Synthetic division is a simplified method for dividing polynomials, specifically for dividing a polynomial by a linear binomial of the form x − c, where c is a constant. It is a faster and more efficient way of performing polynomial division compared to long division, especially when the divisor is a first-degree polynomial.
Steps for Synthetic Division
Below is a step-by-step solution for the division of any linear polynomial with the help of the synthetic division method.
Let’s walk through the synthetic division of dividing: 4x2 - 6x -8 by x - 2 step-by-step.
Step 1: Identifying Coefficients.
The given polynomial is 4x2 − 6x − 8. The coefficients are:
4 -6 -8
There are no missing terms, so no need to insert a zero.
Step 2: Identify the Divisor.
The divisor is x − 2, so set x − 2 = 0, which gives x = 2. Therefore, the value of c in synthetic division is 2.
Write the coefficients in a row and the divisor value on the left:
2 | 4 -6 -8
Step 3: Write the first co-efficient as it is.
Write down the first coefficient as it is, here the coefficient is 4.
Step 4: Multiply and add it with the divisor.
Multiply the first coefficient (4) by the divisor (2). This gives 4 × 2 = 8. Write 8 below the second coefficient, then add it to get the new value.
Step 5: Repeat the step 4.
Continue multiplying the new value by the divisor and adding it to the next coefficient until all coefficients have been processed. Once all coefficients are used, the last value you obtain will be the remainder, and the other values will form the quotient.
Step 6: Derive Result
The result will depend on the last line of the solution given above.
In this case, the quotient is 4x + 2, and the remainder is −4. Therefore, the result of the division is:
\frac{4x^2-6x-8}{x-2}= 4x + 2 - \frac{4}{x-2}
Synthetic Division Vs Long Division
The key differences 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)
The 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, but mistakes can still occur, especially if coefficients are written incorrectly
Synthetic division is a method used to divide a polynomial by a linear factor of the form x − c, where c is a constant.
What is the formula for Synthetic Division?
When dividing a polynomial p(x) by x − a, synthetic division gives a quotient q(x) and a remainder. If a is the root of the divisor, there will be no remainder, making the division exact.
What are the 6 Steps in Synthetic Divison?
The 6 steps in synthetic division are:
Identify the coefficients: Write down the coefficients of the polynomial.
Set the divisor: Set x − a equal to 0, and use aaa in the division.
Write the first coefficient: Bring down the first coefficient as it is.
Multiply and add: Multiply the first coefficient by a, then add it to the next coefficient.
Repeat the process: Continue multiplying and adding for each coefficient.
Derive the result: The final row gives the quotient and remainder.
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.\frac{x^3-5x^2+6x+6}{x-3}= x^2-2x+\frac{6}{x-3}
