An Overview of the Long Division Method of Polynomials
A generalised version of the well-known arithmetic operation known as long division, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree in algebra. The long division method of polynomials is one of the most common methods of dividing polynomials. In this method, there is a divisor, a dividend (which is to be divided), a quotient, and a reminder. This is a very interesting method of dividing polynomials. The image given below describes the different terms used in the long division method of polynomials.
Polynomial Division
What is a Polynomial?
When exponents, constants and variables are combined using mathematical operations like addition, subtraction, multiplication and division, the result is a polynomial. The kind of the polynomial expression may be monomial, binomial or trinomial, depending on how many terms are included in it.
Example: \[9x,2x + 3y,3x + 4z - 9z + 5xyz\] and so on.
Monomial, Binomial and Trinomial
The number of terms in a polynomial determines which of three different sorts of polynomials it is. There are three different kinds of polynomials:
Monomial
Binomial
Trinomial
Types of Polynomials
Monomial
Monomial is an expression that contains only one member. There must be only one term which is nonzero. Here are some examples of monomials:
\[3x,5y\].
Binomial
A binomial expression is a polynomial expression containing exactly two terms. A binomial can be thought of as the sum or difference of two or more monomials. Here are some examples of binomials:
\[3x + 4y,6a + 5b\].
Trinomial
A trinomial is an expression with exactly three terms. Here are some examples of trinomial:
\[2{x^2} + 3x + 4\]
Long Division Method of Polynomial
Polynomial Division is the process of dividing one polynomial into another. You can perform division between different types of polynomials. Between two monomials, between a polynomial and a monomial, or between two polynomials. A polynomial is an n-algebraic expression with expressive variables, terms and coefficients.
Steps of Long Division Method of Polynomial
This method of Long Division Polynomials is as follows:
Sort the terms in descending order of degree.
Write the missing terms using 0 as the coefficient.
For the first term of the quotient, divide the first term of the dividend by the first term of the divisor.
Multiply this quotient by the divisor to get the product.
Subtract this product from the dividend and omit the next section (if any).
The difference and the shortened period form a new dividend.
Follow this process until you get the rest. The remainder can be zero or have an index less than the divisor.
Long Division of Polynomials
Solved Examples on Division Polynomials
1. Solve \[\left( {24{x^2} + 2x + 4} \right) \div \left( {2x + 1} \right)\] using a long division method.
Ans: The image below will give a better understanding on how to divide Polynomials.
Answer: Remainder: 9
Quotient: 12x - 5
2. How are Polynomials Divided?
Ans: In mathematics, there are two ways to divide polynomials. These are the synthetic approach and lengthy division. The long division method is the most challenging and time-consuming to master, as its name suggests. The synthetic technique, on the other hand, is a "fun" way to divide polynomials.
FAQs on Long Division Method of Polynomials
1. What is the long division method of polynomials and in which situations is it commonly applied in Class 10 and 12 Maths as per CBSE 2025-26 syllabus?
The long division method of polynomials is an algorithm used to divide one polynomial by another, especially when the divisor has a lower or equal degree to the dividend. It is commonly applied while simplifying expressions, finding quotients and remainders, and solving algebraic equations as required in the CBSE 2025–26 Maths curriculum.
2. How do you perform the long division of polynomials step by step according to the CBSE prescribed method?
To divide polynomials using the long division method as per CBSE guidelines:
- Arrange both polynomials in descending order of degree.
- Divide the first term of the dividend by the first term of the divisor to find the first quotient term.
- Multiply the divisor by this term and subtract the result from the dividend.
- Repeat the process with the new result until the remainder's degree is less than the divisor's degree.
3. Why is arranging polynomial terms by degree important before starting the long division process?
Arranging polynomial terms in descending order before division is crucial because it ensures each step focuses on the highest-degree terms, preventing mistakes and making the calculation systematic. This method improves clarity and helps in obtaining accurate quotients and remainders.
4. What are the key parts involved in a polynomial division operation?
The main parts in a polynomial division are the dividend (the polynomial being divided), the divisor (the polynomial by which you divide), the quotient (the result), and the remainder (the leftover part, if any).
5. Can the remainder ever be equal to or higher in degree than the divisor when dividing polynomials?
No, using the long division method, the remainder must always have a degree less than the divisor or be zero. If not, further division steps are still required, as highlighted in CBSE exam marking schemes.
6. How do you handle missing terms in a polynomial when using the long division method?
If the given polynomial lacks certain terms, replace them with zero coefficients (e.g., use 0x for missing x term). This maintains the correct place values, preventing errors during division steps as per CBSE recommendation.
7. What is the difference between the long division method and synthetic division for polynomials, and why is long division preferred for CBSE exams?
The long division method is a general procedure suitable for all types of polynomials, including those with higher degrees and non-linear terms, matching CBSE patterns. Synthetic division is a shortcut but only works when the divisor is a linear polynomial (of the form x - a). For board exams, long division is preferred because it is comprehensive and applies to a wider range of problems, as reflected in CBSE's marking guidelines.
8. How can errors during the long division of polynomials affect exam marks and what are some common pitfalls to avoid?
Errors such as not aligning degrees, omitting zero terms, or incorrect subtraction can lead to loss of marks in step-marked questions. To avoid mistakes, follow each step methodically, check calculations after subtraction, and always write missing terms explicitly. This aligns with CBSE's emphasis on method marks.
9. In what types of exam questions is the long division of polynomials frequently asked, and how is it typically marked in CBSE assessments?
Long division of polynomials often appears in 3-mark or 4-mark questions in board exams, where stepwise calculation and correct final answer both carry marks. CBSE marking schemes reward steps like correct arrangement, division, multiplication, subtraction, and identifying the remainder, as per the 2025–26 exam pattern.
10. How do you interpret the quotient and remainder once you have completed a polynomial division question?
The final answer is expressed as: Dividend = Divisor × Quotient + Remainder. The quotient provides the main result of the division, and the remainder shows what's left. If the remainder is zero, the divisor is a factor of the dividend.
11. What happens if the divisor is of a higher degree than the dividend in a polynomial division question?
If the divisor's degree exceeds that of the dividend, the quotient is zero and the remainder is the entire dividend. This situation is sometimes tested to check concept application in board papers.
12. How can understanding the long division method of polynomials help with factorization and solving equations?
Mastering the long division method allows students to test if a polynomial is divisible by another (i.e., if the remainder is zero), which is key in factorization problems and solving higher-degree equations, as required by CBSE curriculum objectives.
