Dividing Polynomials by Monomials

Dividing polynomials by monomials is an algebraic and arithmetic operation where a polynomial is divided by a monomial which is also known as a one-term polynomial. We will be learning different methods of dividing polynomials by monomials in this article.

The most common method that is used to divide polynomials by monomials is by splitting the terms of polynomial separated by (+) or (-) and solving each term separately. The final result will be the combination of all the individual results obtained. Another method of dividing polynomials by monomials is the factorization method. Let us understand both the methods briefly as described below.

Dividing Polynomials by Monomials: Splitting the Terms Method

In this method, we will split the terms of the polynomial separated by the operator (+) or (-) between them and simplify each term. Let's take an example to understand this. For example, (6y² + 3y) / (3y) can be solved by following the steps of dividing polynomials by monomials given below:

• Split the terms of the polynomial 6y² + 3y. The terms are 6y² and 3y.
• Now each term will be divided by the monomial 3y i.e., (6y²/3y) + (3y/3y).
• Each term will be simplified to its lowest form by canceling out the common factors i.e., 6y²/3y = 2y and 3y/3y = 1.
• The individual results are now combined with the (+) operator between them. Therefore the result is 2y + 1.

Hence, we see that the result of (6y² + 3y) / (3y) = 2y + 1.

NOTE: To simplify the division of algebraic terms we simplify the coefficients separately and the variables with powers separately. To simplify 6y²/3y, we first consider the coefficients that are 6/3 = 2, and the variables with exponents will be simplified as y²/y = y. The result is now combined as 2y. Therefore, 6y²/3y = 2y.

Factorization Method of Dividing Polynomials by Monomials

When we divide polynomials by monomials by the factorization method, we first find the common factor between the numerator and the denominator. Let's take an example to understand this. For example, to solve (8x² + 4x) ÷ 4x by factorization method we can follow the below steps of dividing polynomials by monomials:

• We will observe the common factors between the numerator and the denominator. We see that (8x² + 4x) and 4x have a common factor of 4x.
• Thus, the expression can be written by taking the common factor 4x outside the parentheses and the denominator 4x can be retained that is 4x(2x + 1) / 4x.
• Canceling out the common term 4x, we get the result as 2x + 1 as the answer.

Hence, we see that the result of (8x² + 4x) ÷ 4x is 2x + 1.

Let's consider a polynomial a(x) being divided by a monomial b(x) where a(x) is not a multiple of b(x) or b(x) is not a factor of a(x). If b(x) is not a factor of a(x), we end up getting a non-zero remainder when a(x) is divided by b(x). Let's assume the quotient of a(x)/b(x) as q(x) and the remainder as r(x).

We know that, Dividend = Divisor × Quotient + Remainder
i.e., a(x) = b(x) × q(x) + r(x)
a(x) / b(x) = q(x) + r(x) / b(x)

Therefore, the general solution obtained when polynomials are divided by monomials with remainders is in the form q(x) + r(x) / b(x). Let's take an example to understand this.

Example: Divide the polynomial a(x) = 8x³ - 4x² + 2 by b(x) = 4x.
We will use the method of splitting the terms to do the division.
a(x) / b(x) = (8x³ - 4x² + 2) / 4x
= (8x³/4x) - (4x²/4x) + (2/4x)
Now by dividing the coefficients and variables separately we get,
= (2x² - x) + 1/2x

Thus, the value of (8x³ - 4x² + 2) / 4x = (2x² - x) + 1/2x.

Long division of polynomials is an algorithm for dividing polynomial by another polynomial of the same degree or a lower degree. Dividing polynomials by monomials involves the division of a polynomial by a monomial. We will be taking two examples to understand the division of polynomials by monomials using the long division method.

Dividing Polynomial by Monomial without Remainder using Long Division

We will take the example of dividing a polynomial by a monomial without any remainder. Let's divide the polynomial 5x² + 25x by 5x using long division.

Here are the steps to perform the long division of a polynomial by a monomial.

• Step 1: Divide the first term of the dividend (5x²) by the first term of the divisor (5x), and put that as the first term in the quotient (x).
• Step 2: Now, subtract it and bring down the remaining term(s) of the dividend (25x).
• Step 3: Repeat the same process with the new polynomial obtained after subtraction, i.e divide (25x) by (5x) to get the second term of the quotient (5). On subtracting we get 0 which is the remainder.

NOTE: This process is done until the new dividend's degree obtained in every step after subtraction is greater than or equal to the divisor's degree.

Thus, on dividing 5x² + 25x by 5x using long division, we get the quotient as x + 5 and the remainder as 0.

Dividing Polynomial by Monomial with Remainder using Long Division

We will now take an example of a polynomial which is not a multiple of the given monomial and divide them. Note that, since the divisor is not a factor of the dividend, we will get a non-zero remainder. Let's divide 9x² + 15x - 6 by 3x.

The steps remain the same as described above. When we get the new dividend as - 6 here, we stop the division as the degree of - 6 is lesser than the degree of the divisor 3x and therefore the remainder will be - 6.

Thus, on dividing 9x² + 15x - 6 by 3x we get the quotient as 3x + 5 and the remainder as - 6.

Related Topics

Check these articles to know more about the concept of dividing polynomials by monomials.

• Long division of polynomials
• Division Algorithm for Polynomials
• Dividing Two Polynomials
• Division of Polynomial by Linear Factor

How to Divide Polynomials by Monomials?

Polynomials can be divided by monomials by splitting the terms of the polynomial and dividing each of them by the monomial individually and finally connecting them with the operators (+) or (-) according to the sign of the terms. For example, let's divide (5x² - 15x) / 5x.
= (5x²/5x) - (15x/5x)
= x - 3

What are Dividing Polynomials by Monomials Rules?

The rules to divide polynomials by monomials are as follows:

• Split the terms of the polynomial and divide each of them by the monomial independently.
• To solve the division of each term, divide the coefficients separately and the variables with powers separately to get the lowest form of each term.
• Combine the results obtained in the previous step using the operators (+) or (-) as per the sign of the terms.

Example: Divide (12x³ - 6x²) / 3x
= (12x³/3x) - (6x²/3x)
= 4x² - 2x

How to Divide Polynomials by Monomials with Exponents?

Polynomials can be divided by monomials with exponents by first splitting the terms of the polynomial followed by dividing each term by the monomial. While dividing the terms of a polynomial by monomial, we first divide the coefficients separately and then the variable with exponents can be separately divided by using laws of exponents. Finally, the result can be combined. To divide 9a² by 3a, we first divide 9/3 = 3, and a²/a = a, thus the result will be 3a. Let's take an example, divide (27y³ - 18y) / 3y.
= (27y³/3y) - (18y/3y)
= 9y² - 6

What Method of Dividing Polynomials by Monomials is Best for you and Why?

The long division method is the best to divide polynomials by monomials as this includes the divisions which leave remainders as well. Hence, it helps us to get a clear idea about the quotient and the remainder.

How to Divide Polynomials by Monomials using Long Division?

The polynomials are divided by monomials using long division by following the steps below:

• Step 1: Divide the first term of the dividend by the first term of the divisor, and put the result as the first term in the quotient.
• Step 2: Now, subtract it and bring down the rest of the terms.
• Step 3: Repeat the same process with the new polynomial obtained after subtraction that is divide the new dividend by the monomial to get the second term of the quotient.
• Step 4: Repeat the process until the degree of the divisor becomes greater than the degree of the remainder.