Long Division of Polynomials

There are three methods of dividing polynomials: division by monomial method, synthetic division method and the long division method. In cases where the dividend and divisor do not share any common factors, this process helps in effective simplification. Long division of polynomials, similar to long division of numbers, has components like quotient, divisor, dividend, and remainder.
 

While performing long division of polynomials, follow these steps:

Step 1: If required, arrange both polynomials in descending order of powers.

Step 2: Divide the first term of the dividend by the first term of the divisor. This gives you the first term of the quotient.

Step 3: Multiply the first term of the quotient by the divisor. Subtract the product from the dividend.

Step 4: Bring down the next term, if any

Step 5: Repeat the process till the remainder is zero or is of a lower degree than divisor. Then the long division process is complete.
 

There are times when there can be a missing term in the expression while performing long divisions. For example, in the polynomial 2x⁴ + 5x² − 3, the terms x³ and x are missing.
In such cases, either the coefficient is written as zero or a gap is left in place of the coefficient. 
Let's solve the given division problem (2x⁴ + 5x² − 3)  ( x² - 1)

Step 1: Write the missing terms
Dividend: 2x⁴ + 0x³ + 5x² + 0x - 3
Divisor: x² + 0x − 1

Step 2: Divide the leading terms

            
Step 3: Multiply and subtract
Multiply
(x² - 1)  2x² = 2x⁴ - 2x²
Subtract
(2x⁴ + 0x³ + 5x²) - (2x⁴ + 0x³ - 2x²) = 0x⁴ + 0x³ + 7x² 
Bring down: 0x - 3
Divide again: 
              
(x² − 1) × 7 = 7x² − 7
(7x² + 0x − 3) − (7x² + 0x − 7) = 0x² + 0x + 4

The remainder has degree zero, which is less than the divisor's degree of 2.

So the final answer is:

                          
Since the remainder is not zero, (x² - 1) is not a factor. 
 

For dividing polynomials by monomials, note common factors between the numerator and denominator. Then split the expression and divide each term by the monomial individually. Simplify each term individually, then combine them for the final answer.

For example: 

Question: Divide the polynomial:
 (6x³ + 9x² - 12x)  3x
The expression as a fraction: 
    
Simplifying terms:

Combining the simplified terms, we get the answer = 2x² + 3x - 4
 

Another monomial refers to the monomial in the denominator. The process of long division of a polynomial by another monomial is similar to the process mentioned above. List the prime factors of the numerator and denominator and cancel out all common factors.

For example, divide the polynomial

  
We split and simplify each term,

 3x²y² + 2xy
 

When there are no common factors between the numerator and denominator, long division by binomials is used. 
Let's understand the steps involved through an example:

Question: Divide (4x² + 8x + 5) by ( x + 2)

Step 1: Divide the first term of the dividend by the divisor 

       
Step 2: Multiply the divisor by 4x 
    ( x + 2 ) (4x) = 4x² + 8x

Step 3: Subtracting from the dividend gives a new polynomial
   (4x² + 8x + 5) - (4x² + 8x) = 0x² + 0x + 5 = 5 

Step 4: Bring down the remainder and repeat the process.
While dividing a polynomial (4x² + 8x + 5) by a binomial (x + 2), the quotient is 4x and the remainder is 5.
 

Long division of polynomials by other polynomials can be compared to the long division process of numbers. The process is similar to the above-mentioned process of long division of polynomials. Let's understand each step using an example:

Question: Divide the polynomials 8x³  + 10x²  − 6x + 5 ÷ x² + 2x + 1

Solution: 
Step 1: Divide the first term of the dividend by the divisor

           
Step 2: Multiply the divisor by 8x

   (8x) (x² + 2x + 1) = 8x³ + 16x² + 8x

Step 3: Subtract by dividend 

       (8x³ + 10x² - 6x + 5) - (8x³ + 16x² + 8x)

 (0x³ - 6x² - 14x + 5)

Step 4: Repeat the process.
 
The next term of the quotient is -6

 (−6) (x² + 2x + 1) = −6x² −12x −6

 (−6x² − 14x + 5) − (−6x² − 12x −6)

 0x² − 2x + 11

dividing 8x³ + 10x² − 6x + 5 ÷ x² + 2x + 1, gives us quotient = 8x - 6 and remainder = -2x + 11

Long division of polynomials simplifies complex expressions. It helps break down functions for easier analysis.

Simplifying physical phenomena like projectile motion, electric circuits, or wave functions 

Long division helps simplify polynomial functions used in analyzing complex physical systems.

Coding theory

Polynomial division is used in error detection and correction in coding. In cryptography, long division helps simplify expressions and find remainders in modular arithmetic.

Designing mechanical gear

Long division simplifies motion equations. It also helps determine relationships between moving parts.

Find economic trends

Polynomial functions are used to model cost, revenue, and profit over time and production units. The long division method helps find trends, break-even points or marginal costs and profits. 

Example: 
Cost function: C(x) =4x³ + 2x² + 5x + 10
Revenue function: R(x) = 6x³ + 3x² +7x
Profit function: P(x) = R(x) − C(x)

Predicting planetary motion

Long division simplifies polynomial functions, calculating orbital paths to interpretable forms.