Dividing Polynomials

Polynomials are algebraic expressions that consist of variables and constants. Polynomials can be written in the form: ax2 + bx + c, arranged in descending order of their degree.  

Division is one of the basic arithmetic operations, where a number is broken down into equal parts. Dividing polynomials includes dividing a polynomial by a monomial or a binomial. For example, when dividing 2x² + 4x + 24 by 2x + 12, it can be written as: 

2x² + 4x + 12/2x + 12

Here, the numerator is 2x² + 4x + 12 and the denominator is 2x + 12. That means the numerator becomes the dividend and the denominator becomes the divisor

Both polynomials and monomials are algebraic expressions, but the polynomials have multiple terms, whereas the monomials have only one term. To divide a polynomial by a monomial, there are two ways: 

• Splitting the terms method 
   
• Factorization method. 

In the splitting the term method, the terms of the polynomials are split by the operations between them, and then each term is separately divided by the divisor. For example, 22x² + 12 by 2x

Splitting the term: 22x² + 12 

The terms are 22x² and 12

Dividing each term by the divisor:

22x² / 2x = 11x

12/2x = 6/x

22x² + 12 / 2x = 11x + (6/x)

In the factorization method, we find the common factor between the numerator and denominator of the polynomial by factoring the polynomial. For example, when dividing 22x² + 12x by 2x

The common factor from 22x² + 12x is 2x(11x + 6)

Now it can be expressed as: 2x(11x + 6) / 2x

Cancelling out the common factors, here the common factor is 2x

So, (2x(11x + 6)) / 2x = 11x + 6

To divide polynomials by binomials, we use the long division and synthetic division methods. We use these methods when the polynomials won’t share a common factor. 

The long division method is used to divide a polynomial by another polynomial. So, both the dividend and divisor have two or more terms. Follow these steps to divide polynomials using long division, using an example:

Step 1: Dividing the first term of the dividend by the first term of the divisor. The result is the first term of the quotient. 

For example, when dividing 3x² + 8x + 4 by x + 2

Dividing the first terms: 3x² / x = 3x

So, here, 3x is the first term in the quotient. 

Step 2: Multiply the divisor by the answer in step 1, and write below the dividend 

Here we multiply (x + 2) by 3x, that is 3x(x + 2) = 3x² + 6x

Step 3: Subtract the new polynomial from the dividend 
So, subtracting (3x² + 8x + 4) - (3x² + 6x) = 2x + 4

Step 4: The process is repeated with the same polynomial

Divide: 2x / x = 2

Add +2 as the quotient

Multiplying (x + 2) by 2, 2x + 4

Subtracting (2x + 4) - (2x + 4) = 0

So, the quotient is 3x + 2.

The synthetic division is the method used to divide polynomials by a binomial of the form x - k. Here, the focus is on the coefficient, which makes this process quicker and easier. Follow these steps for dividing polynomials using the synthetic division:

For example, dividing x² + 5x + 6 by x - 2

Step 1: Find the value of k and write it on the left side

To find the value of k, we first write the divisor in the form x - k. 
Here, the divisor is x - 2, so k = 2

Step 2: Writing the coefficients of the dividend on the right of K

The dividend is: x2 + 5x + 6

So, the coefficients are: 1, 5, 6

The coefficients are written on the right and k on the left. 

Step 3: Bring down the coefficient

Bringing down the coefficient of the highest degree term of the dividend, here it is 1. 

Step 4: Multiply and add

Now we multiply the k by the first coefficient and write the product below the second coefficient, and add them.

Here, the value of k is 2 and the first coefficient is 1, so 2 × 1 = 2

Adding 5 + 2 = 7

Step 5: The process is repeated

Now we multiply k by the second coefficient obtained in step 4.

Here, multiply 2 and 7, 2 × 7 = 14

Write 14 below 6 and add them; 6 + 14 = 20

Step 6: The final answer will be one degree less than the dividend. For example, if the dividend has x² then the quotient will be x. 

Here, the highest degree of dividend is x², so the quotient's higher degree would be x. The quotient is x + 7, and the remainder is 20. 

So, x² + 5x + 6/x - 2 = x + 7 + 20/x - 2

The division of polynomials is used in different fields such as engineering, computer graphics, economics, civil engineering, and so on. Here are some applications of dividing polynomials. 

• In structural engineering, polynomial equations are used to understand the behavior of structures. 

• In certain cryptography algorithms, we use polynomial division.

• To model economic trends and predict the future in economics, we use polynomial division. 

• In signal processing for filtering and noise reduction, polynomials are used.