Division of Polynomial by Another Monomial: Definition, Types, Methods

A polynomial is an algebraic expression. It consists of variables and coefficients. Dividing polynomials is an algorithm for solving rational numbers that are polynomials divided by monomials or other polynomials. Polynomial division follows the same rules that apply to integer division.

Key Terms: Polynomials, Monomials, Algebraic expression, Binomial, Polynomial division, Coefficients, Integer

Also read: Isosceles Triangle Theorems

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Definition of Polynomials

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An algebraic expression that consists of a variable and a constant with an integer exponent of the variable is known as a Polynomial. 

Below is an example of Polynomial algebraic expression: 

Here,

• 3x³ is term 
• x is variable 
• 2, 3 (raised to the power) are exponents
• The term before variables (3,5,7) are coefficient 
• Number 6 is constant 

Polynomials 

The video below explains this:

Polynomials Detailed Video Explanation:

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Types of Polynomials

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Polynomials are segregated according to the number of terms and their degree. There are two types of classification.

• Based on the number of terms
• Based on the Degree of a Polynomial

Based on Number of Terms

Polynomials are divided into three types based on the number of terms: monomials, binomials, and trinomials.

1. Monomial: A monomial is a polynomial type with only one term. For Example 5, 2x, 5a², 6xy⁵, 2x, 3a², 8xy
2. Binomial: A binomial is a polynomial in which two terms are separated by an addition (+) or subtraction (-) sign. Examples of binomials are 5x+3, 9x–1, 8x+5y, 6x–4y
3. Trinomial: A polynomial with exactly three terms is called a trinomial. An example of a trinomial is 8x³- 4x²+8x+7, 3x+9x²–5x³ 

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Based on Degree

Based on the degree, polynomials can be classified into zero or constant polynomials, linear polynomials, cubic polynomials, quadratic polynomials, fourth-order polynomials, and so on. 

1. Constant or zero polynomial: A constant or zero polynomial whose power of a variable is zero. If the power of the variable is zero, then variable x * 0 = 1, so its value is 1. Zero polynomials have terms that are constants like 3, 2, 6, 10.
2. Linear polynomials: Polynomials of which the maximum power of a variable or the degree of the polynomial is 1 are linear polynomials. For Example, x–2, y+2, a+3x–1.
3. Quadratic polynomial: A polynomial whose maximum power of a variable or degree of the polynomial is 2 is a quadratic polynomial. For Example x²+x, y²+3, a²-4x²+x, y²+1, a²+6, etc.
4. Cubic Polynomials: Polynomials of which the maximum power of a variable or the degree of the polynomial is 3 are cubic polynomials. For example y³+5, x³–2, 15+a³, x+2y³+8
5. 4th-degree/Quartic polynomial: A polynomial with a variable or polynomial with a maximum degree of 4 is called a 4th or 4th-degree polynomial. For Example x⁴+4x³–2x²-x+1, 7y⁴–y²+1,3y⁴–y²+1, etc.

Polynomials Classification Based on Degree

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Polynomial Operations

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There are four primary polynomial operations: 

• Adding Polynomials- Polynomials of the same degree and variables are added. The result is also a polynomial
• Polynomial Subtraction- Polynomials of the same degree and variable are subtracted. The result is also a polynomial 
• Polynomial Multiplication- Multiplying two or more polynomials always results in a higher-order polynomial.
• Polynomial Division- Gives a remainder (fraction form)

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Methods of Dividing Polynomial by Another Monomial

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The division is the splitting of quantities into equal parts. In mathematics, repeating subtraction or the inverse of multiplication is called division. For example, if you divide 30 by 3, you get 10 because 3 is subtracted 10 times from 30.

When dividing a polynomial by a monomial, the division can be performed in two ways. One is to separate the operations Addition (+) and subtraction (-) to solve each part individually. Another approach is to do simple factorization and further simplification.

Splitting Method 

Polynomial terms are divided by a monomial to simplify.

To divide the polynomial by the monomial, put the polynomial as Dividend and monomial as the divisor. Find the common denominator. Arrange the terms of the Dividend in the descending order of their power/index. Now, divide each term of the polynomial with the monomial. 

For example 

• 6m⁵ + 2m⁴ - 8m² ÷ 2m²

Solution:

 6m⁵ + 2m⁴ - 8m² 

= -------------------

2m²

Separate each term

= (6m⁵/2m²) + (2m⁴/2m²) - (8m²/2m²)

= 3m³ + 1m² - 4

• 4x³y + 20x²y² - 2xy ÷ 2xy

Solution :

4x³y + 10x²y² - 2xy

= --------------------

2xy

= (4x³y/2xy) + (10x²y²/2xy) - (2xy/2xy)

= 2x² + 5xy - 1

Factorization Method

When dividing a polynomial by Factorization method, find a common factor between the numerator and the denominator.

For example 

• (4x² + 8x) ÷ 4x. 

The numerator and denominator have a common factor of 4x.

Therefore, the algebraic expression is 4x(x + 2) / 4x. Now Cancel out the common term 4x, answer is x+2. 

• x³+2x²+20x/ x²+10x

=x(x²+2x+20) / x(x+10).

x²+2x+20 / x+10…..(cancel out the common factor x)

=x²+2x+20 x+10.

=(x+10) (x+5) x+2.

Now it is apparent the numerator and denominator have a common factor of x+2.

Therefore the answer is x+2.

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Solved Examples

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1. x² + 6y² / 3x²y

Solution:

= x² + 6y² / 3x²y

Dividing each term with 3x²y

= (x² / 3x²y) + (6y² / 3x²y)

Simplifying the equation 

= (1 / 3y) + (2y / x²)

2. Divide the polynomial a² + ab – ac by the monomial –a

Solution: 

= a²+ab−ac/−a

Dividing each term with −a

= a² /−a+ ab/−a− ac/−a

= − a² /−a− ab/a+ ac/a

After simplifying we get the equation 

= -a - b + c

3. Divide the polynomial 4m⁴n⁴ – 8m³n⁴ + 6mn³ by the monomial -2mn

Solution:

= 4m⁴n⁴ – 8m³n⁴ + 6mn³ ÷ -2mn

= 4m⁴n⁴ – 8m³n⁴ + 6mn³ ÷ -2mn

= 4m⁴n⁴ – 8m³n⁴ + 6mn³−2mn

Dividing each term with the monomial -2mn

= 4m⁴n⁴/−2mn− 8m³n⁴/−2mn+ 6mn³/−2mn

= −4m⁴n⁴/2mn+ 8m³n⁴/2mn− 6mn³/2mn

After simplifying we get the equation 

= 2m³n³ + 4m²n³ - 3n²

4. Divide the polynomial (5x²−6x) by the monomial 3x 

Solution:

=(5x²−6x) /3x 

Find the common factor between the polynomial (5x²−6x) and the monomial 3x 

=x(5x−6)/3x

After cutting the common factor, we get the equation

=(5x−6)/3 

5. Divide the polynomial (x³+2x²+3x) by the monomial 2x

Solution:

= (x³+2x²+3x)​/ 2x

Finding the common factor 

=x(x³+2x+3)​/ 2x

Cutting the common factor to get the answer 

= (x²+2x+3)/​2

6. Divide the polynomial 12x³ – 15xy + 6x by the monomial 3x.

Solution:

=(12x³–15xy + 6x) / 3x

Dividing each term with the monomial 3x

=(12x³/3x) – (15xy/3x) + (6x/3x)

After simplifying the algebraic expression, we get 

= 4x² – 5y + 2 

7. x + 4x² + 12x⁴ - x⁵ ÷ by 2x

Solution :

First arrange all the terms in decreasing order of exponents

-x5 + 12x⁴ + 4x² + x ÷ 4x

 -x5 + 12x⁴ + 4x² + x

= --------------------

4x

Dividing each term with the monomial 4x

= (-x⁵/4x) + (12x⁴/4x) + (4x²/4x) + (x/4x)

After simplifying the algebraic expression, we get

= (-1/4) x⁴ + 3x³ + x + ¼

8. Divide 10x³y + 2x²y² – 20xy by 2xy

Solution:

(10x³y + 2x²y² – 20xy) /(2xy) 

Dividing each term with the monomial 2xy

= 10x³y /2xy + 2x²y²/2xy – 20xy/2xy

After simplifying the algebraic expression, we get 

= 5x² + xy – 10

Also Read:

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Things to Remember

• An algebraic expression that consists of a variable and a constant with an integer exponent of the variable is known as a Polynomial. 
• There are two types of polynomials: Based on the number of terms and Based on the Degree of a Polynomial
• The division of polynomial by a monomial can be done in two ways: Splitting Method and Simple factorization
• To divide a polynomial by a monomial, divide each term by the monomial. (signs are important)
• A monomial is a polynomial type with only one term. (4, 2x, 6a², 5xy⁵, 8x, 2a², 3xy)

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