There are two different methods to divide the polynomials in Mathematics. One is the long division method and the second one is the synthetic division method. Among these two methods, the synthetic division method is the shortcut method to divide polynomials. It is also known as the polynomial division method of a special case when it is divided by the linear factor. The synthetic division replaces the long division method. In certain situations, you will find this method easier. In this article, we will discuss what synthetic division method is and how to perform synthetic division method along with steps with
What is the Synthetic Division of Polynomials?
The Synthetic division of polynomials is a shortcut of polynomial division, especially if we need to divide by a linear factor.
Synthetic division method is basically used to find out the zeros or roots of polynomials and this method is not for the division of factors. Thus, the formal definition of synthetic division can be defined as:
“Synthetic division of polynomials can be defined as a simplified way of dividing a polynomial with another polynomial equation of degree 1 and this method is generally used to find the zeros of polynomials”
This division method (synthetic division method of polynomials) is performed manually with less effort of calculation than the long division method which requires a lot of calculation. Usually, a binomial term is used as a divisor in this division method, such as x – c.
Synthetic Division Steps (synthetic division method of polynomials)-
Here are synthetic method examples for better understanding!
Divide x2+5x +6 by x-1 using the concept of synthetic division.
Let’s go step by step,
Synthetic Division Steps-
Now, solve the same question. Comparing, here through the synthetic division of polynomial steps can see that you got the same result from the synthetic division, the same quotient (that is, 1x + 6) and the same remainder at the end (which is, 12). You will get the same answer if you perform long division for the same question.
The results we have got are formatted differently, but we should recognize that each format provided us with the result, being a quotient of x + 6, we always get a remainder of 12.
Why the Synthetic Division Method?
Synthetic division is generally known as a shorthand, or shortcut, a method of polynomial division in the special case of dividing by a linear factor and synthetic division works only in this case. Synthetic division method is generally used, however, not for dividing out factors but for finding zeros or roots of polynomials. The method is easier than the long division method for solving synthetic division problems and that’s why it is preferred.
FAQs on Synthetic Division of Polynomials
1. What is meant by synthetic division of polynomials as per the CBSE Class 12 Maths syllabus?
Synthetic division is an efficient method for dividing a polynomial by a linear binomial of the form x - a. It simplifies the long division process into a sequence of operations using only the coefficients, helping students quickly find the quotient and remainder.
2. What are the step-by-step procedures to solve a polynomial using the synthetic division method?
To perform synthetic division as per CBSE 2025-26 guidelines, follow these steps:
- Write coefficients of the dividend polynomial in order. If a power is missing, use zero.
- Set the divisor x - a equal to zero and solve for a.
- Place a outside the setup; bring down the leading coefficient.
- Multiply the number below by a and add to next coefficient; repeat till the end.
- The final row gives you the coefficients of the quotient and the last number as remainder.
3. Why is synthetic division preferred over long division in certain cases?
Synthetic division is often preferred for dividing polynomials by linear factors (x - a) because it is quicker and involves fewer steps than long division. This method reduces calculation errors and simplifies the work, especially in examination settings where time management is crucial.
4. What types of errors should students avoid when applying the synthetic division method in exams?
Students should:
- Ensure every term of the polynomial is represented (use zeros for any missing degrees).
- Apply synthetic division only when dividing by a linear binomial (x - a).
- Check calculations at each step to prevent cascading arithmetic mistakes.
5. How do you interpret the quotient and remainder obtained after performing synthetic division?
The results from synthetic division give:
- Quotient: The polynomial of one degree less than the original.
- Remainder: The final value, which can be written as a fraction over the divisor if required—that is, P(x) = (x - a) × Q(x) + R.
6. How can synthetic division assist in finding the zeroes of a polynomial efficiently?
Synthetic division helps test possible roots quickly. If the remainder is zero after dividing by x - a, then a is a zero of the polynomial. This method aids in polynomial factorization and root-finding as per the CBSE syllabus.
7. Can synthetic division be applied when dividing by non-linear factors or higher-degree polynomials?
Synthetic division is designed specifically for linear divisors (x - a). It cannot be used for quadratic or higher-degree divisors. For such cases, polynomial long division is the recommended method.
8. Who developed the synthetic division method and what is its historical significance in mathematics?
The method was developed by Paolo Ruffini in the early 19th century. Synthetic division contributed to the evolution of algebra by streamlining solutions for polynomial equations and is now a standard tool in high school mathematics curricula such as CBSE and NCERT.
9. In what ways does synthetic division help when solving higher-order equations in CBSE board exams?
Synthetic division allows for rapid simplification and checking of potential roots, making it a valuable exam strategy when solving cubic or higher-degree polynomial equations, especially for questions requiring factorization or remainder calculation.
10. What are common misconceptions students have about synthetic division in board exams?
Some common misconceptions include:
- Using synthetic division for non-linear divisors.
- Neglecting to fill in missing terms with zeros in the setup.
- Confusing the remainder position among the results.
- Not interpreting the quotient’s variable powers correctly.
