Long Division Calculator| Definition, Steps, and Examples

What is Long Division?

Long division is a method for dividing larger numbers (or algebraic expressions) into smaller, manageable steps, breaking division into repeated subtraction and multiplication. This is especially useful when the dividend is too large for mental math.  

In symbolic form, division is written as $\frac{a}{b}$, where:

• $a$ = dividend (the number/expression being divided),
• $b$ = divisor (the number/expression doing the dividing),
• The result = quotient. 

For example:

• Dividing 125 by 5 using long division gives a quotient of 25, since $125 \div 5 = 25$.
• In algebra, polynomial long division simplifies expressions like dividing $x^2 + 3x + 2$ by $x + 1$.

Students often practice long division manually, but a long division calculator can instantly verify steps and results.

How to Calculate Long Division Step-by-Step

1、Write the dividend inside the division bracket and the divisor outside.

2、Compare the divisor to the leftmost digits of the dividend to find how many times the divisor fits.

3、Multiply the divisor by the quotient digit, then subtract the result from the current portion of the dividend.

4、Bring down the next digit of the dividend and repeat steps 2–3 until no digits remain.

5、If the divisor does not divide evenly, record the remaining value as the remainder.

Examples of Long Division

Example 1: Basic Long Division (No Remainder)

Divide 144 by 12.

1、Place 144 inside the division bracket and 12 outside.

2、Calculate $12 \times 12 = 144$, so the quotient is 12.

3、Subtract $144 - 144 = 0$ (no remainder).

Example 2: Long Division with Remainder

Divide 50 by 6.

1、Place 50 inside the division bracket and 6 outside.

2、Calculate $6 \times 8 = 48$, so the quotient is 8.

3、Subtract $50 - 48 = 2$ (remainder = 2).

Example 3: Polynomial Long Division

Divide $x^2 + 3x + 2$ by $x + 1$.

1、Divide the leading term of the dividend ($x^2$) by the leading term of the divisor ($x$) to get $x$.

• Multiply: $x(x + 1) = x^2 + x$.
• Subtract: $(x^2 + 3x + 2) - (x^2 + x) = 2x + 2$.

2、Divide the new leading term ($2x$) by $x$ to get 2.

• Multiply: $2(x + 1) = 2x + 2$.
• Subtract: $(2x + 2) - (2x + 2) = 0$ (no remainder).

3、Final quotient: $x + 2$.  

Related Practice Problems

Problem 1: Arithmetic Long Division

A school has 120 students and wants to divide them into groups of 15. How many groups are needed?

Solution:

1、Dividend = 120, divisor = 15.

2、Perform division: $120 \div 15 = 8$.

3、Verify: $8 \times 15 = 120$.

Answer: 8 groups.

Problem 2: Polynomial Long Division

Divide $2x^3 + 3x^2 - x - 5$ by $x - 1$.

Solution:

1、Divide $2x^3$ by $x$ to get $2x^2$.

• Multiply: $2x^2(x - 1) = 2x^3 - 2x^2$.
• Subtract: $(2x^3 + 3x^2 - x - 5) - (2x^3 - 2x^2) = 5x^2 - x - 5$.

2、Divide $5x^2$ by $x$ to get $5x$.

• Multiply: $5x(x - 1) = 5x^2 - 5x$.
• Subtract: $(5x^2 - x - 5) - (5x^2 - 5x) = 4x - 5$.

3、Divide $4x$ by $x$ to get 4.

• Multiply: $4(x - 1) = 4x - 4$.
• Subtract: $(4x - 5) - (4x - 4) = -1$ (remainder = -1).

Answer: Quotient = $2x^2 + 5x + 4$, Remainder = $-1$.

Long division applies to both arithmetic and algebra—manual practice builds foundational skills, while a long division calculator ensures accuracy and efficiency.

Common Mistakes

A frequent mistake in long division is confusing the dividend and the divisor, which can change the meaning of the problem entirely. Another common error is forgetting to account for the remainder when the dividend does not divide evenly by the divisor. In algebra, students may misalign terms during polynomial long division, leading to incorrect quotients. Using a long division calculator can help confirm results and highlight where errors may occur.

Solving Tips

When working through long division, always double-check by multiplying the quotient by the divisor to see if it equals the dividend. Pay careful attention to each subtraction step to avoid miscalculations. For algebraic expressions, align terms correctly during polynomial long division to maintain accuracy. If you are using a long division calculator or an AI math solver, make sure the inputs are entered precisely. Finally, practice regularly—consistent problem solving is the best way to build speed and accuracy.

Why is long division important?

Long division plays a critical role in both academic study and real-world problem solving. Here are some key reasons why it is important:

• In Mathematics: Long division is a fundamental operation for understanding fractions, ratios, and algebra. It also provides the foundation for advanced concepts like polynomial long division, which is essential in simplifying expressions and solving equations. Students often verify these steps with a long division calculator to ensure accuracy.
• In Science and Engineering: Division is applied in calculating densities, speeds, and rates. Engineers use it to analyze loads and stresses in structures, ensuring safety and efficiency.
• In Data Analysis: Division helps compute averages, percentages, and growth rates, all of which are central to interpreting data and making informed decisions.
• In Everyday Life: Long division is practical in budgeting, cooking, and planning. Whether dividing expenses among friends or calculating the cost per item, it simplifies daily tasks and resource management.

Use GoMim Math AI Solver for long division

Step 1: Visit the GoMim AI website and select the AI Math Solver option.

Step 2: Enter the long division problem you need to solve in the input field.

Step 3: Click the 'Solve' button to receive a step-by-step solution to your division problem.

Step 4: Review the solution and use the steps to understand the process better.

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FAQ

Q: What is division in math terms?

A: Division is a mathematical operation where a number, called the dividend, is divided by another number, the divisor, to find a quotient. It is often expressed as \(\frac{a}{b}\) or \(a \div b\).

Q: Why is division used?

A: Division is used to distribute a number into equal parts, to find how many times one number is contained within another, and to solve problems involving fractions, ratios, and rates.

Q: What are some examples of division in real life?

A: Real-life examples of division include calculating the cost per item, dividing resources evenly, or determining the number of people each portion can serve when sharing food.

Q: Can division have a remainder?

A: Yes, division can have a remainder when the dividend is not perfectly divisible by the divisor, leaving a remainder that is less than the divisor.

Q: How do you check division results?

A: You can check division results by multiplying the quotient by the divisor and adding any remainder. The result should equal the original dividend.

Q: How is division related to multiplication?

A: Division is the inverse operation of multiplication. While multiplication combines equal groups, division breaks a number into equal parts or groups.

Q: What is long division?

A: Long division is a method of dividing larger numbers by breaking down the process into simpler steps, making it easier to handle complex divisions manually.