Long Division of Polynomials
Long division of polynomials is the process of dividing one polynomial with another. Division can be done among the different types of polynomials i.e. between two monomials, a polynomial and a monomial, or between two polynomials. A polynomial is n algebraic expression with variables, terms, and coefficients with the degree of the expressions. Let us explore the division of polynomials by learning about the methods to divide using long division, long division with polynomials, long division with missing terms, the algorithm, and solved a few examples to understand the process better.
What is Long Division of Polynomials?
A long division polynomial is an algorithm for dividing polynomial by another polynomial of the same or a lower degree. The long division of polynomials also consists of the divisor, quotient, dividend, and the remainder as in the long division method of numbers. Observe the numerator and denominator in the long division of polynomials as shown in the figure.
The long division of polynomials also consists of a divisor, a quotient, a dividend, and a remainder.
In algebra, the division of algebraic expressions can be done in three ways:
 Division of a monomial by another monomial.
 Division of a polynomial by a monomial.
 Division of a polynomial by a binomial.
 Division of a polynomial by another polynomial.
Steps For Long Division of Polynomials
The following are the steps for the long division of polynomials:
 Step 1. Arrange the terms in the decreasing order of their indices (if required). Write the missing terms with zero as their coefficient.
 Step 2. For the first term of the quotient, divide the first term of the dividend by the first term of the divisor.
 Step 3. Multiply this term of the quotient by the divisor to get the product.
 Step 4. Subtract this product from the dividend, and bring down the next term (if any). The difference and the brought down term will form the new dividend.
 Step 5. Follow this process until you get a remainder, which can be zero or of a lower index than the divisor.
Long Division of Polynomial by Missing Terms
While performing long division of polynomials, there can be a missing term in the expression, for example, 6x^{4} + 3x  9x^{2} + 6, x^{3} is missing. In this case, we either leave a gap while dividing or we write the coefficient as zero. Let's understand how to do the long division of polynomials with the same example. We need to divide the polynomial a(x) = 6x^{4} + 3x  9x^{2} + 6 by the quadratic polynomial b(x) = x^{2}  2
Arrange the polynomial in the descending order of the power of the variable.
a(x) = 6x^{4}  9x^{2} + 3x + 6
b(x) = x^{2}  2
Divide a(x) by b(x) in the same way as we divide numbers.
Add the missing indices with zero (0) as the coefficient.
Divide 6x^{4} by x^{2} to get the first term of the quotient. We get 6x^{2}.
Multiply the divisor by 6x^{2}.
Divide 3x^{2} by x^{2} to get the next term of the quotient.
As the power of the next dividend is less than the divisor, we get our required remainder. Please remember that as the remainder we got is a nonzero term, we can say that x^{2}  2 is not a factor of 6x^{4}  9x^{2} + 3x + 6. Therefore, the quotient is 6x^{2} + 3 and the remainder is 3x.
Long Division of Polynomials by Monomials
While dividing polynomials by monomials, write the common factor between the numerator and the denominator of the polynomial and divide each term separately. Once the result is obtained, add all the terms together to form an expression. For example: Divide the following polynomial: (2x^{2} + 4x + 8xy) ÷ 2x. Both the numerator and denominator have a common factor of 2x. Thus, the expression can be written as 2x(x + 2 + 4y) / 2x. Canceling out the common term 2x, we get x + 4y + 2 as the answer.
Long Division of Polynomials by Other Monomial
Long division of polynomials by another monomial is done in a similar manner as done for polynomials by monomials. The factors of the monomial of both the numerator and denominator are listed out and the long division takes place. For example, divide 62x^{3} by 2x. The factors of 62x^{3} = 2 × 31 × x × x × x and 2x = 2 × x. The common factors for both are 2x. Hence, 62x^{3}/2x = 31x^{2}.
Long Division of Polynomials by Binomials
Long division of polynomials by binomials is done when there are no common factors between the numerator and the denominator, or if you can't find the factors. Let us go through the algorithm of dividing polynomials by binomials using an example: Divide: (6x^{2}  4x  24) ÷ (x  3). Here, (6x^{2}  4x  24) is the dividend, and (x  3) is the divisor which is a binomial. Observe the division shown below, followed by the steps.
 Step 1. Divide the first term of the dividend (6x^{2}) by the first term of the divisor (x), and put that as the first term in the quotient (6x).
 Step 2. Multiply the divisor by that answer, place the product (6x^{2}  18x) below the dividend.
 Step 3. Subtract to create a new polynomial (14x  24).
 Step 4. Repeat the same process with the new polynomial obtained after subtraction.
So, when we are dividing a polynomial (6x^{2}  4x  24) with a binomial (x  3), the quotient is 6x + 14 and the remainder is 18.
Long Division of Polynomials by Other Polynomial
Long division of a polynomial with another polynomial is done when the expression is written in the standard form i.e. the terms of the dividend and the divisor are arranged in decreasing order of their degrees. The long division method for polynomials is considered the generalized version of the simple long division method done with numbers. Let us look at an example to understand this better. The process of division is very similar to the rest of the methods. Divide the polynomial 6x^{3} + 12x^{2} + 2x + 25 by x^{2} + 4x + 3. Here, 6x^{3} + 12x^{2} + 2x + 25 is the dividend, and x^{2} + 4x + 3 is the divisor which is also a polynomial.
 Step 1: Divide the first term of the dividend (6x^{3}) by the first term of the divisor (x^{2}), and put that as the first term in the quotient (6x).
 Step 2: Multiply the divisor by that answer, place the product (6x^{3} + 24x^{2} + 18x) below the dividend.
 Step 3: Subtract to create a new polynomial (12x^{2}  16x + 25).
 Step 4: Repeat the same process with the new polynomial obtained after subtraction.
So, when we are dividing a polynomial 6x^{3} + 12x^{2} + 2x + 25 with a binomial x^{2} + 4x + 3, the quotient is 6x  12 and the remainder is 32x + 61.
Long Division Algorithm of Polynomials
The division algorithm for polynomials says, if p(x) and g(x) are the two polynomials, where g(x) ≠ 0, we can write the division of polynomials as: p(x) = q(x) × g(x) + r(x).
Where,
 p(x) is the dividend.
 q(x) is the quotient.
 g(x) is the divisor.
 r(x) is the remainder.
 r(x) = 0 or degree of r(x) < degree of g(x)
If we compare this to the regular division of numbers, we can easily understand this as: Dividend = (Divisor X Quotient) + Remainder. Let us take the previous example,
p(x) = 6x^{3} + 12x^{2} + 2x + 25
g(x) = x^{2} + 4x + 3
q(x) = 6x  12
r(x) = 32x + 61
Apply the division algorithm, q(x) × g(x) + r(x)
(6x  12) × (x^{2} + 4x + 3) + (32x + 61)
6x^{3} + 24x^{2} + 18x  12x^{2}  48x  36 + 32x + 61
6x^{3} + 12x^{2}  2x + 25
= p(x).
Hence, the division algorithm is verified.
Related Topics
Check these articles to know more about the concept of dividing polynomials and its related topics.
Polynomials Long Division Examples

Example 1: Rose wants to divide the polynomial 4x^{3}  3x^{2} + 4x by 2x+1. Can you help her with the solution?
Solution: Here, the polynomial 4x^{3}  3x^{2} + 4x is divided by 2x+1
Therefore, \(\text{quotient = } 2x \dfrac{3}{2} \text{ and remainder = } x+ \dfrac32\).

Example 2: Solve (24a^{2} + 48a+2) ÷ (6a + 12) by using the method of long division of polynomials.
Solution: The long division of (24a^{2} + 48a+2) ÷ (6a + 12) can be done in the following way.
Therefore, quotient = 4a and remainder = 2.

Example 3: Consider the following two polynomials: a(x) = x^{3}  x^{2} + x  1 and b(x) = 2x + 1.
Find the quotient polynomial and the remainder when a(x) is divided by b(x).
Solution: We proceed as earlier:
\(\therefore \)\[\begin{align}&q\left( x \right)=\frac{1}{2}{x^2}  \frac{3}{4}x + \frac{7}{8}\\&r =  \frac{{15}}{8}\end{align}\]
FAQs on Long Division of Polynomials
What is the Meaning of Long Division of Polynomials?
Long division of polynomials is a technique followed in Algebra to divide a polynomial by another polynomial of a lower or the same degree.
How Do You Divide Polynomials by Long Division?
The following are the steps for the long division of polynomials:
 Arrange the terms in the decreasing order of their indices (if required). Write the missing terms with zero as their coefficient.
 For the first term of the quotient, divide the first term of the dividend by the first term of the divisor.
 Multiply this term of the quotient by the divisor to get the product.
 Subtract this product from the dividend, and bring down the next term (if any). The difference and the brought down term will form the new dividend.
 Follow this process until you get a remainder, which can be zero or of a lower index than the divisor.
What is the Importance of Long Division of Polynomials?
Long division of polynomials is a way to test whether one polynomial has the other one as a factor. It also helps in breaking the dividend into a simple sequence by easy steps.
What are the Advantages of Long Division of Polynomials?
The advantage of long division of polynomials is that it is a simple and widely used method to divide two polynomials in less space and requires lesser calculations.
What are the Disadvantages of Long Division of Polynomials?
The only disadvantage of long division of polynomials is that in case the divisor is nonlinear, the calculations become more complex.
What are the Main Uses of a Synthetic Division of Polynomials?
The main use of synthetic division of polynomials is that it is used when the divisor is linear and the coefficient of the variable in it is one.
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