In the field of mathematics, Polynomial Long Division Method is the procedure used to represent the action of dividing any two polynomials by one another. It means to divide two polynomials with the same number of terms, two polynomials of different kinds, or a polynomial and a monomial as one method to execute this.

A polynomial is a mathematical expression that combines the addition, subtraction, multiplication, and division of four fundamental mathematical operations with coefficients, constant terms, and variables.

## The general expression for Polynomial:

In algebra, the term polynomial is generally defined as an expression written in the form of a_{n}x^{n}+a_{n-1}x^{n-1}+…………………a_{2}x^{2} + a_{1}x + a_{o}, where “n” may be zero or positive integer, “a” is any coefficient term, and “x” is any variable term.

The highest exponential power of “x” in the above expression is called the degree of the polynomial.

## What is polynomial long division?

This is the process of dividing a polynomial by another of the same or different degree. A simple division process in which we have to eliminate the first term of the dividend polynomial by multiplying the first term of the divisor with a suitable term does this.

This division of two polynomials consists of different terms like;

- The divisor (an expression by which other expressions be divided)
- The quotient (a required solution to this division)
- The dividend (an expression that has to be divided)
- The remainder (the term remained at the end of the division process)

## Steps for Long Division of Polynomials

For the long division of polynomials, the following steps are taken:

- To arrange all the terms of polynomials from greater to lower i-e; in descending order.
- To write the missing term of any exponent as zero of that term like 0x
^{2 }(if any) - Multiply a suitable number, variable, or both with the first term of the divisor term so that the first term of the dividend is eliminated and we find the first term of the quotient.
- Subtract the similar terms and write the next term down to the second step. The subtraction process will give us the new term as a dividend
- Write the term down as the remainder if any of the terms is not similar to another.
- Continue this process until we find a reminder of the given polynomial, which may be zero or the term having a degree less than the divisor.

## How to perform long division of polynomials?

You can find the long division of polynomials with the help of a polynomial long division calculator to get the results in fraction of seconds. For manual calculations, below are a few solved examples.

**Example 1: **Find quotient and remainder, when (x^{4 }+ 2x^{3 }− x^{2 }+ 5x – 8) is divided by (x + 4)

**Solution: **

Step 1: Division process

x^{3 }– 2x^{2}+ 7x –23 | |

x +4 | x^{4}+2x^{3}−x^{2}+5x–8
x |

– 2x^{3} – x^{2}
– 2x | |

7x^{2 }+ 5x
7x | |

–23x –8
–23x – 92 | |

84 |

Here,

**Divisor= **x +4

**Dividend= **x^{4}+2x^{3}−x^{2}+5x–8

**Quotient= **x^{3 }– 2x^{2}+ 7x –23

**Remainder= **84

**Example 2: **For what value of K, is the given divisor (x+6) the root of (5x^{3} + 12x^{2}+53x+K)

**Solution:**

Step 1: Division process

5x^{2}–18x–55 | |

x +6 | 5x^{3} + 12x^{2}+53x+K
5x |

–18x^{2}+53x + K
–18x | |

–55x + K
–55x –330 | |

K+330 |

For checking (x+6) as a root, we have

K+330=0

⇒ k= –330

**Example 3: **Find quotient and remainder, when (5x^{4}+12x^{3}−2x^{2}+15x–8) is divided by (x+10)

**Solution: **

Step 1: Division process

5x^{3 }– 38x^{2}+ 378x –3765 | |

x +10 | 5x^{4}+12x^{3}−2x^{2}+15x–8
5x |

– 38x^{3} –2x^{2}
– 38x | |

378x^{2 }+ 15x
378x | |

–3765x –8
–3765x – 37650 | |

37642 |

Here,

**Divisor= **x +10

**Dividend= **5x^{4}+12x^{3}−2x^{2}+15x–8

**Quotient= **5x^{3 }– 38x^{2}+ 378x –3765

**Remainder= **37642

**Example 4: **Find the quotient when the polynomial (2x^{3}+2x+5) is divided by (x +1)

**Solution:**

Step 1: To write the dividend in descending order of exponents and write as zero for the missing term just like the above dividend as 2**x ^{3}+0x^{2}+2x+5**

Step 2: Division process

2x^{2} – 2x + 4 | |

x+ 1 | 2x^{3}+ 0x^{2 }+ 2x + 5
2x |

– 2x^{2} + 2x + 5
– 2x | |

4x + 5
4x + 4 | |

1 |

Here,

**Divisor= **x+1

**Dividend= **2x^{3}+2x+5

**Quotient= **2x^{2} – 2x + 4

**Remainder=** 8

## Applications of Polynomial long division in Engineering:

The idea of polynomial long division is frequently employed in engineering. Here are some instances of how polynomial long division is frequently used to solve and comprehend engineering-related challenges.

**In electrical engineering:**

The majority of electrical engineers utilize the polynomial long division approach in some complicated circuits to develop a special way to transfer the current flow and assess the stability of the circuits by modeling circuit components and signals as polynomials.

**In communication systems engineering:**

Polynomial long division provides a helping hand to detect an error we face ever. It also helps us in encoding and decoding signals.

The majority of electrical engineers utilize the polynomial long division approach in some complicated circuits to develop a special way to transfer the current flow and assess the stability of circuits by modeling circuit components and signals as polynomials.

**In Interpolation:**

Polynomial long division plays a vital role in the approximation of different functions to obtain the most suitable data that best fit on the given inputs. Through this, we can find a unique solution with fewer possibilities of errors and a stage in the interpolation procedure which is closely connected to interpolation.

The process of estimating or creating a function that traverses a collection of provided data points is referred to as interpolation. Finding a polynomial of degree n that passes through the n+1 provided data points is the goal of polynomial interpolation.

## Conclusion

In this above discussion, we have learned the concept of Polynomial Long Division, its general expression, and different terms use in writing a polynomial. Through its measuring steps, we can easily understand the way of completing a problem in a sophisticated manner.

Different kinds of examples will help us to solve this division process in different directions. The information about its usage in everyday life reminds us of its importance. After understanding the whole concept, we can solve all sorts of problems relevant to it.