Dividing Polynomials
The division of polynomials is an algorithm to solve a rational number which represents a polynomial divided by a monomial or another polynomial. The division of two polynomials may or may not result in a polynomial. In this chapter, we will learn the concept of dividing polynomials, which is slightly more detailed than multiplying them. We will also learn how to divide polynomials with binomials and other polynomials.
1.  What is Dividing Polynomials? 
2.  Simple Division 
3.  Long Division of Polynomials 
What is Dividing Polynomials?
Polynomials are algebraic expressions that consist of variables and coefficients. It is written in the following format: 5x^{2} + 6x  17. This polynomial has three terms that are arranged according to their degree. The term with the highest degree is placed first, followed by the lower ones. The division of polynomials is an algorithm to solve a rational number that represents a polynomial divided by a monomial or another polynomial. The divisor and the dividend is placed exactly the same way as we do for regular division. For example, if we need to divide 5x^{2} + 7x + 25 by 6x  25, we write it in this way:
\[\dfrac{(5 x ^2+7 x+25)}{(6 x 25)}\]
The polynomial written on top of the bar is the numerator ( 5x^{2} + 7x + 25), while the polynomial written below the bar is the denominator (6x  25). This can be understood by the following figure which shows that the numerator becomes the dividend and the denominator becomes the divisor:
Simple Division
While dividing a polynomial with a monomial, the division can be done in two ways. One is by simply separating the '+' and '' operator signs. That means, we break the polynomial from the operating sign, and solve each part separately. Another method is to do the simple factorization and further simplifying. Let us have a look at both the methods in detail:
Splitting the Terms
Split the terms of the polynomial separated by the operator ( '+' or '' ) between them and simplify each term. For example, (4x^{2}  6x) ÷ (2x) can be solved as shown here. We first take common terms from the numerators and denominators of both the terms, we get, [(4x^{2}) / (2x)]  [(6x) / (2x)]. Cancelling the common term 2x from the numerator and the denominator, we get 2x  3.
Factorization
When you divide polynomials you may have to factor your polynomials to find a common factor between the numerator and the denominator. For example: Divide the following polynomial: (2x^{2} + 4x) ÷ (x + 2)
Both the numerator and denominator have a common factor of (x+2). Thus, the expression can be written as: 2x(x + 2) / (x + 2)
Canceling out the common term x + 2, we get, 2x.
Long Division of Polynomials
When there are no common factors between the numerator and the denominator, or if you can't find the factors, you can use the long division process to simplify the expression.
Long Division Without Remainder
Let us go through the algorithm for the long division of polynomials using an example: Divide: (4x^{2}  5x  21) ÷ (x  3). Here, (4x^{2}  5x  21) is the dividend and (x  3) is the divisor. Observe the division shown below, followed by the steps.
Step 1. Divide the first term of the dividend (4x^{2}) by the first term of the divisor (x), and put that as the first term in the quotient (4x).
Step 2. Multiply the divisor by that answer, place the product (4x^{2}  12x) below the dividend.
Step 3. Subtract to create a new polynomial (7x  21).
Step 4. Repeat the same process with the new polynomial obtained after subtraction.
Long Division With Remainder
The long division of a polynomial with a remainder follows the same steps as that with the remainder. Here is an example, where (7x^{2} + 35x + 24) needs to be divided by (x + 4). Observe the division shown below which gives the quotient 7x +7 and the remainder 4.
Solved Examples on Dividing Polynomials

Example 1: Alex is stuck on a problem while working on polynomial division. Can you help him solve the following to obtain the quotient:
(x^{4}  10x^{3} + 27x^{2}  46x + 28) ÷ (x  7)
Solution:
Therefore, the quotient is x^{3}  3x^{2 }+ 6x  4.

Example 2: Stacy needs help in finding the remainder for the following division of polynomials. Can you solve this?
(4x^{3} + 5x^{2} + 5x + 8) ÷ (4x + 1)
Solution:
Therefore, the remainder is 7.
FAQs on Dividing Polynomials
Why is Dividing Polynomials Important?
Dividing Polynomials is important because it provides an algorithm to solve a rational number that represents a polynomial divided by a monomial or another polynomial.
What is the Easiest Way to Divide Polynomials?
The easiest way to divide polynomials is by using the long division method. However, in the case of the division of polynomials by a monomial, it can be directly solved by splitting the terms or by factorization.
How do you Divide Polynomials with Two Variables?
The division of polynomials with two variables can be done using the long division method or the synthetic method of division of polynomials by binomials. When there are no common factors between the numerator and the denominator or if you can't find the factors, we can use the long division method or the synthetic method to simplify the expression.
How do you Know if Polynomials are Dividing?
If there is no remainder left after the completion of the division process, then it means that the polynomials are dividing completely. If a remainder is left, then the polynomial is not divisible.
What is Polynomial Division Used for in RealLife?
We use polynomial division for various aspects of our daytoday lives. We need it for coding, engineering, designing, architecting, and various other reallife areas.
How do you Divide Polynomials?
The easiest way to divide polynomials is by using the long division method. However, in the case of the division of polynomials by a monomial, it can be directly solved by splitting the terms or by factorization.