Division Method

Division: The arithmetic operation includes the four essential functions, and they are addition, subtraction, division and multiplication on all the types of real numbers, including the integers.

The division symbol is in the form of an obelus as a horizontal line with a dot above and below the line, \( \div .\) It was first used as the sign for the division by the Swiss mathematician Johann Rahn in his book Teutsche Algebra in \(1659.\) In this article, we will discuss everything about the division operation, different methods and division algorithm along with solved examples.

What is Division?

Definition: The term division is one of the four basic arithmetic operations, out of which the other three are addition, subtraction and multiplication. Division can be defined as the splitting of the more prominent groups into small groups equally.

Example: If you want to divide the numbers \(30 \div 6,\) in this \(30\) is the dividend, the number \(6\) is the divisor, and the result you get is \(30 \div 6 = 5,\) the number \(5\) is the quotient.

Division of Work

The division of work is the course of tasks assigned to and completed by a group of workers to increase efficiency. Division of labour, also known as division of labour, is the breaking down of a job to have several different tasks that make up the whole.

Long Division Method

The term long division is the method for dividing the large numbers into steps sequence wise. It is the most common method used to calculate the problems based on division. You can see the given diagram of long division to understand the divisor, the dividend, the quotient, and the remainder.

Parts of Long Division

You can see the terms related to the long division, which can also be said as parts of long division that are used in the regular division:

Dividend: This is the number that has to be divided.
Divisor: These are the numbers that will divide the dividend.
Quotient: This is the result you get after division.
Remainder: This is the leftover part that cannot be divided further.

Process of Long Division

The term division is one of the four basic arithmetic operations, out of which the other three are addition, subtraction and multiplication. The long division is the standard division algorithm used to divide the large numbers by breaking down the division problem into more straightforward steps.

The divisor is split from the dividend by the right parenthesis \(\left\langle ) \right\rangle \) or vertical bar \(\left\langle |\right\rangle ,\) and the dividend is set apart from the quotient by the overbar. So, look at the given steps below to understand the process of long division:

1. The first step is you have to take the first digit of the given dividend. Then you have to verify if this digit is greater than or equal to the given divisor.
2. You have to divide the digit by the given divisor and then write the top as the quotient.
3. Here, you will subtract the answer from the digit and then write the difference below.
4. After that, you have to bring down the next digit (if present).
5. You have to repeat the same process until you divide the whole number.

Floor Division

The term floor division means a regular division operation, except it returns the biggest possible integer. The integer is either less than or equal to the usual division answer that you get.

Example: Dividing the two integers \(101\) by \(4.\) In this, \(100\) is the numerator, and the number \(4\) is the denominator.
Now, the integer division \(\frac{{101}}{4}\) gives you the number \(25\) along with the remainder \(1.\)

Division Algorithm

Definition: When we divide \(3{x^2}\) by \(x,\) we get
\(\frac{{3{x^2}}}{x} = 3x,\) here dividend \( = 3{x^2},\) divisor\( = x,\) quotient\( = 3x\) and remainder\( = 0\)
So, \(3{x^2} = x \times 3x + 0\)
Here, the division algorithm for the polynomials can be written as shown below:
\({\text{Dividend}} = \left({{\text{Divisor}} \times {\text{Quotient}}} \right) + {\text{Remainder}}\)
Usually, if \(p\left( x \right)\) and \(g\left( x \right)\) are the two polynomials and that degree of \(p\left( x \right) \ge \) degree of \(g\left( x \right)\) and \(g\left( x \right) \ne 0,\) then you can find the polynomials \(q\left( x \right)\) and \(r\left( x \right)\) such that:
\(p\left( x \right) = g\left( x \right)q\left( x \right) + r\left( x \right).\)
Here, \(r\left( x \right) = 0\) or the degree of \(r\left( x \right) < \) degree of \(g\left( x \right).\) We can say that \(p\left( x \right)\) divided by \(g\left( x \right),\) gives \(q\left( x \right)\) as the quotient and \(r\left( x \right)\) as the remainder.

Example: Find the quotient of \(\left({{x^3} – 8{x^2} + 19x – 12} \right) \div \left({x – 1} \right.)\)
First, you will write the dividend which you are dividing, \({x^3} – 8{x^2} + 19x – 12,\) inside the long division bar and outside, you have to write the divisor, dividing by \(x – 1.\)
You will start with the most significant power terms \(x\) and \({x^3}.\)
Here, write the term with which you will multiply \(x\) to generate \({x^3}:{x^2}\)
Now, multiply the divisor by the term \(\left({{x^2}} \right)\) and place the result below the dividend, so the like terms are aligned.
Later subtract the new polynomial from the real dividend. Write down the rest of the words to acquire the latest dividend

After multiplying both the terms in the divisor by the number \(12,\) arrange the like terms and then subtract where everything is cancelled and get no remainder, i.e., zero.
Hence, in this case, we obtain quotient \( = {x^2} – 7x + 12\) and remainder \( = 0.\)

In general, if \(f\left( x \right)\) and \(g\left( x \right)\) are two polynomials with \(g\left( x \right) \ne 0,\) then we can find polynomials \(q\left( x \right)\) and \(r\left( x \right)\) such that:
\(f\left( x \right) = g\left( x \right) \cdot q\left( x \right) + r\left( x \right),\) where \(r\left( x \right) = 0\) or \(\deg r\left( x \right) < \deg g\left( x \right)\)
The above process is known as the division algorithm of polynomials.
If \(r\left( x \right) = 0,\) then \(g\left( x \right)\) is called a factor of \(f\left( x \right).\)
If \(f\left( x \right) \ge \deg \,g\left( x \right),\) then \(\deg q\left( x \right) = \deg f\left( x \right) – \deg g\left( x \right).\)

Division Using Factorisation Using Algebraic Identity

In this method, we use standard algebraic identities to factorise either or both the polynomials given as dividend and divisor and cancel the common factors from both numerator and the denominator.

Example: We can divide \({x^3} – 8\) by \(x – 2\) as follows:
\(\frac{{{x^3} – 8}}{{x – 2}} = \frac{{{{\left( x \right)}^3} – {{\left( 2 \right)}^3}}}{{x – 2}} = \frac{{\left({x – 2}\right)\left[{{{\left( x \right)}^2} + x \times 2 + {{\left( x \right)}^2}} \right]}}{{x – 2}} = \frac{{\left({x – 2} \right)\left[{{x^2} + 2x + 4} \right]}}{{\left({x – 2} \right)}}\)
\( = {x^2} + 2x + 4.\)
Here, the algebraic identity \({a^3} – {b^3} = \left({a – b} \right)\left({{a^2} + ab + {b^2}} \right)\) is used.

Solved Examples – Division

Q.1. Find the number which, when divided by \(46\) gives a quotient \(11\) and remainder \(18.\)
Ans: We have,
Divisor \( = 46,\) Quotient \( = 11\) and Remainder \( = 18.\)
We have to find the dividend. By division algorithm we have,
\({\text{Dividend}} = {\text{Divisor}} \times {\text{Quotient}} + {\text{Remainder}}\)
\( \Rightarrow \)Dividend \( = 46 \times 11 + 18\)
\( = 506 + 18 = 524.\)
Hence the required answer is \(524.\)

Q.2. Find the value of: \(\left[{32 + 2 \times 17 + \left({ – 6} \right)} \right] \div 15\)
Ans: We have, \(\left[{32 + 2 \times 17 + \left({ – 6} \right)} \right] \div 15\)
\( = \left[{32 + 34 + \left({ – 6} \right)} \right] \div 15 = \left({66 – 6} \right) \div 15 = 60 \div 15 = \frac{{60}}{{15}} = 4\)
Hence, the required answer is \(4.\)

Q.3. Divide the numbers \(7528 \div 5\) using the long division method.
Ans: We have, \(7528 \div 5\)
The number \(7528\) is the dividend, and the number \(5\) is the divisor.

Here, you have got the quotient as \(1505\) and the remainder as \(3.\)
Hence, the required answer is shown above.

Q.4. Divide the polynomial \(f\left( x \right) = 14{x^3} – 5{x^2} + 9x – 1\) by the polynomial \(g\left( x \right) = 2x – 1.\) Also, find the quotient and remainder and verify the division algorithm.
Ans: Using the long division method, we obtain

Quotient \(q\left( x \right) = 7{x^2} + x + 5\) and remainder \(r\left( x \right) = 4.\)
Now, \(q\left( x \right)g\left( x\right) + r\left( x \right) = \left({7{x^2} + x + 5} \right)\left({2x – 1}\right) + 4\) \( = 14{x^3} + 2{x^2} + 10x – 7{x^2} – x – 5 + 4\)
\( = 14{x^3} – 5{x^2} + 9x – 1\)
\( = f\left( x \right)\)
i.e. \(f\left( x \right) = g\left( x \right)q\left( x \right) + r\left( x\right)\) or \({\text{Dividend}} = {\text{Quotient}} \times {\text{Divisor}} + {\text{Remainder}}.\)

Q.5. Divide: \({x^{4a}} + {x^{2a}}{y^{2b}} + {y^{4b}}\) by \({x^{2a}} + {x^a}{y^b} + {y^{2b}}\)
Ans: We have,
\({x^{4a}} + {x^{2a}}{y^{2b}} + {y^{4b}}\)
\( = {\left({{x^{2a}}} \right)^2} + {x^{2a}}{y^{2b}} + {\left({{y^{2b}}} \right)^2}\)
\( = {\left({{x^{2a}}} \right)^2} + 2{x^{2a}}{y^{2b}} + {\left({{y^{2b}}} \right)^2} – {x^{2a}}{y^{2b}}\)
\( = {\left({{x^{2a}} + {y^{2b}}} \right)^2} – {\left({{x^a}{y^b}} \right)^2}\)
\( = \left({{x^{2a}} + {y^{2b}} + {x^a}{y^b}} \right)\left({{x^{2a}} + {y^{2b}} – {x^a}{y^b}}\right)\)
\( = \left({{x^{2a}} + {x^a}{y^b} + {y^{2b}}}\right)\left({{x^{2a}} – {x^a}{y^b} + {y^{2b}}} \right)\)
\(\frac{{{x^{4a}} + {x^{2a}}{y^{2b}} + {y^{4b}}}}{{{x^{2a}} + {x^a}{y^b} + {y^{2b}}}} = \frac{{\left( {{x^{2a}} + {x^a}{y^b} + {y^{2b}}}\right)\left({{x^{2a}} – {x^a}{y^b} + {y^{2b}}} \right)}}{{{x^{2a}} + {x^a}{y^b} + {y^{2b}}}} = {x^{2a}} – {x^a}{y^b} + {y^{2b}}\)

Summary

In the given article, you have looked at the definition of the division, then discussed the division of work. Also, we discussed the long division method that included parts and the process of the long division. We glanced at the floor division and then division algorithm followed by division using factorising using the algebraic identity. Then, we have provided few solved examples along with a few FAQs.

Frequently Asked Questions

Q.1. What are the parts of division?
Ans: The types of the division are as below:
Dividend – The number which is to be divided by another number.
Divisor – The number by which another number is to be divided.
Quotient – The result which you get after the division.
Remainder – The leftover number which cannot be divided further.

Q.2. How do you do the division?
Ans: To divide any numbers, you have to follow the below-given steps:
1. The first step is you have to take the first digit of the given dividend. Then you have to verify if this digit is greater than or equal to the given divisor.
2. You have to divide the digit by the given divisor and then write the top as the quotient.
3. Here, you will subtract the answer from the digit and then write the difference below.
4. After that, you have to bring down the next digit (if present).
5. You have to repeat the same process until you divide the whole number.

Q.3. What is division?
Ans: The term division is one of the four basic arithmetic operations, out of which the other three are addition, subtraction and multiplication. Division can be defined as the breaking of the more prominent groups into small groups equally.

Example: If you want to divide the numbers \(45 \div 5,\) in this \(45\) is the dividend, the number \(5\) is the divisor, and the result you get is \(45 \div 5 = 9,\) the number \(9\) is the quotient.

Q.4. How to divide in excel?
Ans: To divide the numbers in excel, you have to type the equal to a symbol \(\left( = \right)\) in the cell. Then you have to type the number you want to divide, i.e., dividend, followed by a forward slash \(\left( / \right)\) and write the number, i.e., divisor and press on the enter button to calculate the formula.
Example: To divide the number \(35\) by the number \(7\) you type the given expression in the cell: \( = \frac{{35}}{7}\)

Q.5. What are the three forms of division?
Ans: The three forms of the division are given below:
(i) Dividend
(ii) Divisor and
(iii) Quotient

We hope this detailed article on division helped you in your studies. If you have any doubts or queries regarding this topic, feel to ask us in the comment section. We will be more than happy to assist you. Happy learning!