Long Division of Polynomials
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Last updated on December 17, 2025

We use the long division method to divide polynomials, where the dividend and divisor have no common factors. It is a frequently used method among all types of polynomials. A polynomial is an algebraic expression made of variables, coefficients, and terms. Long division can be used between pairs of monomials, polynomials, or between a monomial and a polynomial.

What is the Long Division of Polynomials?

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There are three methods of dividing polynomials: division by monomial method, synthetic division method and the long division method. In cases where the dividend and divisor do not share any common factors, this process helps in effective simplification. Long division of polynomials, similar to long division of numbers, has components like quotient, divisor, dividend, and remainder.
 

What are the Steps for Long Division of Polynomials?

While performing long division of polynomials, follow these steps:


Step 1: If required, arrange both polynomials in descending order of powers.


Step 2: Divide the first term of the dividend by the first term of the divisor. This gives you the first term of the quotient.


Step 3: Multiply the first term of the quotient by the divisor. Subtract the product from the dividend.


Step 4: Bring down the next term, if any


Step 5: Repeat the process till the remainder is zero or is of a lower degree than divisor. Then the long division process is complete.
 

Long Division of Polynomial by Missing Terms

There are times when there can be a missing term in the expression while performing long divisions.
For example, in the polynomial 2x4 + 5x2 − 3, the terms x3 and x are missing.
In such cases, either the coefficient is written as zero or a gap is left in place of the coefficient. 

Let's solve the given division problem (2x4 + 5x2 − 3) ÷ (x2 - 1)


Step 1: Write the missing terms
Dividend: 2x4 + 0x3 + 5x2 + 0x - 3
Divisor: x2 + 0x − 1


Step 2: Divide the leading terms

\( \frac{2x^4}{x^2} = 2x^2 \)

             Step 3: Multiply and subtract

Multiply: (x2 - 1)  2x2 = 2x4 - 2x2
Subtract: (2x4 + 0x3 + 5x2) - (2x4 + 0x3 - 2x2) = 0x4 + 0x3 + 7x2

Step 4: Bring down: 0x - 3

Divide again: \({7x^2 \over x^2} = 7\)             
(x2 − 1) × 7 = 7x2 − 7
(7x2 + 0x − 3) − (7x2 + 0x − 7) = 0x2 + 0x + 4

The remainder has degree zero, which is less than the divisor's degree of 2.


So the final answer is: \(2x^2 + 7 + {4 \over x^2 - 1}\)


Since the remainder is not zero, (x2 - 1) is not a factor. 
 

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Long Division of Polynomials by Monomials

For dividing polynomials by monomials, note common factors between the numerator and denominator. Then split the expression and divide each term by the monomial individually. Simplify each term individually, then combine them for the final answer.

For example: Divide the polynomial:  (6x3 + 9x2 - 12x) ÷ 3x

Step 1: The expression as a fraction: \(\frac{6x^3 + 9x^2 - 12x} {{3x}}\)

     Step 2: Simplifying terms: 

  • \({6x^3 \over 3x} = 2x^2\)
     
  • \({9x^2 \over 3x} = 3x\)
     
  • \({12x \over 3x} = 4x\)
     

Step 3: Combining the simplified terms, we get the answer = 2x2 + 3x - 4

Long Division of Polynomials by Other Monomial

Here, the denominator is another monomial. The process of long division of a polynomial by another monomial is similar to the process mentioned above. List the prime factors of the numerator and denominator and cancel out all common factors.

For example, divide the polynomial 45x4y2 + 30x2y by 15x2

   We split and simplify each term,


\({{45x^4 y^2} \over {15x^2}} + {{30x^3y} \over {15x^2}} \)

\( {{45x^4 y^2} \over {15x^2}} = {3x^2y^2} \)

\( {{30x^3y} \over {15x^2}} = {2xy}\)


Combining the terms: 3x2y2 + 2xy
 

Long Division of Polynomials by Binomials

When there are no common factors between the numerator and denominator, long division by binomials is used. 
Let's understand the steps involved through an example:


Question: Divide (4x2 + 8x + 5) by ( x + 2)


Step 1: Divide the first term of the dividend by the divisor 
\({4x^2 \over x} = 4x \)

       
Step 2: Multiply the divisor by 4x 
 (x + 2) (4x) = 4x2 + 8x


Step 3: Subtracting from the dividend gives a new polynomial
   (4x2 + 8x + 5) - (4x2 + 8x) = 0x2 + 0x + 5 = 5 


Step 4: Bring down the next term, then continue the division process until complete.
While dividing a polynomial, 4x² + 8x + 5 ÷ x + 2 → quotient 4x, remainder 5

Long Division of Polynomials by Other Polynomial

Long division of polynomials by other polynomials can be compared to the long division process of numbers. The process is similar to the above-mentioned process of long division of polynomials. Let's understand each step using an example:


Question: Divide the polynomials 8x3  + 10x2  − 6x + 5 ÷ x2 + 2x + 1


Solution: 
Step 1: Divide the first term of the dividend by the divisor

\({8x^3 \over x^2} = 8x\)
           


Step 2: Multiply the divisor by 8x

   (8x) (x2 + 2x + 1) = 8x3 + 16x2 + 8x

Step 3: Subtract by dividend 

       (8x3 + 10x2 - 6x + 5) - (8x3 + 16x2 + 8x)

   ⇒ (0x3 - 6x2 - 14x + 5)

Step 4: Repeat the process.
 \({{-6x^2} \over x^2} = -6\)

The next term of the quotient is -6

⇒ (−6) (x2 + 2x + 1) = −6x2 −12x −6

⇒ (−6x2 − 14x + 5) − (−6x2 − 12x −6)

⇒ 0x2 − 2x + 11

Dividing 8x3 + 10x2 − 6x + 5 ÷ x2 + 2x + 1, gives us quotient = 8x - 6 and remainder = -2x + 11

Tips and Tricks to Master Long Division of Polynomials

Long Division of Polynomials is a complex mathematical concept which can be quite difficult to understand. Therefore, mentioned below are some tips and tricks to master the topic.

  • Always Arrange in Standard Form: Before dividing, write both the dividend and divisor in descending order of powers of 𝑥.
     
  • Divide Only the Leading Terms: At each step, divide the first term of the dividend by the first term of the divisor and not the entire expression.
     
  • Multiply Carefully: Always multiply the entire divisor by the new quotient term, using parentheses to avoid sign errors.
     
  • Subtract with Attention to Signs: When subtracting, change every sign in the expression being subtracted.
     
  • Bring Down the Next Term: After subtraction, bring down the next term from the dividend before continuing.
     

  • Use Simple Examples First: Teachers can ask students to begin with lower-degree polynomials and gradually increase the difficulty level.
     

  • Using the Division Algorithm: Parents can ask students to verify their solution by substituting it into the division algorithm: Dividend = (Divisor × Quotient) + Remainder.
     

  • Encourage Step-by-Step Work: Teachers can encourage students to write each step clearly and neatly. This helps students to avoid confusion and reduce careless mistakes.

Common Mistakes and How to Avoid Them in Long Division of Polynomials

When performing long division of polynomials, it is possible to make errors. Some things to be aware of, to avoid incorrect results are:

Real-Life Applications of Long Division of Polynomials

Long division of polynomials simplifies complex expressions. It helps break down functions for easier analysis.

  • Simplifying physical phenomena: Long division helps simplify polynomial functions used in analyzing complex physical systems like projectile motion, electric circuits, or wave functions. ​​​
  • Coding theory: Polynomial division is used in error detection and correction in coding. In cryptography, long division helps simplify expressions and find remainders in modular arithmetic.
  • Designing mechanical gear: Long division simplifies motion equations. It also helps determine relationships between moving parts.
  • Find economic trends: Polynomial functions model cost, revenue, and profit over time and across production units. The long division method helps find trends, break-even points or marginal costs and profits. 
  • Predicting planetary motion: Long division simplifies polynomial functions, calculating orbital paths to interpretable forms.

Problem 1

Divide: (6x² + 8x) ÷ 2x

Okay, lets begin

Quotient = 3x + 4, remainder = 0.
 

Explanation

\({ 6x^2 \over 2x} = 3x\)

\({8x \over 2x} = 4\)

Well explained 👍

Problem 2

Divide (9x^3− 6x² + 3x) ÷ 3x

Okay, lets begin

Quotient = 3x2 − 2x + 1, remainder = 0
 

Explanation

Break the expression into separate terms:
\({9x^3 \over 3x}  - {6x^2 \over 3x} + {3x \over 3x}\)

Now simplify each term:

\(9x^3\over 3x\) = 3x2

\(6x^2\over 3x\)  = 2x

\(3x\over 3x \)= 1 

Well explained 👍

Problem 3

Divide: (3x cube + 7x² − x + 2) ÷ (x + 2)

Okay, lets begin

Quotient = 3x2 + x − 3, Remainder = 8.
 

Explanation

3x3 ÷ x = 3x2

Multiply: 3x2 (x + 2) = 3x3 + 6x2

Subtract: (3x3 + 7x2) − (3x3 + 6x2) = x2

Bring down -x

x2 ÷ x = x

Multiply: x(x + 2) = x2 + 2x

Subtract: (x2 − x) − (x2 + 2x) = −3x

Bring down +2
−3x ÷ x = −3

Multiply: −3(x + 2) = −3x − 6

Subtract: (−3x + 2) − (−3x − 6 ) = 8

  3x2 + x - 3 + \(8\over x+2\)

Well explained 👍

Problem 4

Divide (5x^3 + 3x² − 7x + 4) ÷ (x² − x + 2)

Okay, lets begin

\({{5x^3 + 3x^2 - 7x + 4} \over {x^2 - x + 2}} = {5x + 8 - {9x + 12 \over x^2 - x + 2}}\)

Explanation

Dividend: 5x3 + 3x2 − 7x + 4

Divisor: x2 − x + 2

Dividing leading terms: \(5x^3\over 5x^2\) = 5x

Multiply the divisor by 5x: 5x(x2 − x + 2) = 5x3 − 5x2 + 10x

Subtract: (5x3 + 3x2 − 7x + 4) − (5x3 − 5x2 + 10x) 

= (5x3 − 5x3) + (3x2 +5x2) + (−7x − 10x) + 4 

= 0 + 8x2 − 17x + 4

Now divide 8x2 ÷ x2 = 8

Multiply divisor by 8: 8(x2 − x + 2) = 8x2 − 8x + 16

Subtract: (8x2 − 17x + 4) − (8x2 − 8x + 16) 

= (8x2 − 8x2) + (−17x + 8x) + (4 − 16)

= − 9x − 12

Well explained 👍

Problem 5

Divide (2x^4 − 3x^3 + x² + 5) ÷ (x² − 2)

Okay, lets begin

 Quotient: 2x2 − 3x + 5, Remainder = −6x + 15.
 

Explanation

Dividing the leading terms, we get -3x

Multiply the divisor:
−3x(x2 − 2) = −3x3 + 6x

Subtract:
(−3x3 + 5x2 + 5) − (−3x3 + 6x) = 0 + 5x2 −6x + 5

Well explained 👍

FAQs on Long Division of Polynomials

1.How to divide polynomials with long division?

To divide polynomials with long division, follow these steps:

  • Set up the division, and write it in a long division format.
     
  • Divide the first term of the dividend by the first term of the divisor
     
  • Multiply
     
  • Subtract
     
  • Repeat
     

Continue this process till the remainder is zero or of a lower degree than the divisor.
 

2.What do you do if a term is missing in the dividend (e.g., no x² term)?

One can insert a placeholder with zero as its coefficient value.
 

3.When should you use long division instead of synthetic division?

Use long division when the divisor is not in the form x - a. For example: Divide f(x) = 2x3 + 3x2 − x + 4 by x 2 + 1. Since the divisor is degree 2, one must use a long division. Synthetic division is used when the divisor is linear and in the form x - a.
 

4.What is the importance of long division of polynomials?

Long division is important for the foundation of algebra and calculus. It helps simplify complex algebraic fractions, find quotients and remainders, verify factors, and apply the remainder theorem.

5.What are the advantages and disadvantages of long division? Answer:

Advantages

Disadvantages

It is applicable for any divisor, unlike synthetic division.

Time-consuming method

It is easy to apply and follow.

Prone to frequent errors

Helps simplify complex rational expressions.

Not always as efficient as synthetic division

It is useful for calculus functions, like simplifying rational functions for limits.

Calculations become longer when the  divisor is nonlinear

Builds groundwork for higher mathematics

Doesn't offer much visual insight like graphing

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.