A complete lesson on long division

Long division is an important algorithm for division in mathematics, especially for multi-digit division and polynomial division. By mastering the steps and techniques of long division, you will not only improve your mathematical skills, but will also be able to meet the challenges of the DSE Mathematics Examination. This tutorial systematically introduces the basic concepts, calculation methods and application techniques of long division to help you master this important mathematical tool.

Students are learning the mathematical process of long division.

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Basic Concepts - Definition and Principles of the Long Division Law

Long Division, also known as Straight Division, is an important algorithm in mathematics that deals with the operation of division, especially for multi-digit division and polynomial division. It is an important algorithm in mathematics that deals with division operations, especially for multi-digit division and polynomial division. It is a systematic procedure that breaks down a complex division problem into simpler operations.

Basic Principles of Long Division

Long division is based on the following basic relationship:

Divisor = Divisor × Quotient + Remainder

In long division, we solve division problems by following these steps:

  • Breaks down complex division into several simple subtraction and multiplication steps.
  • Divide bit by bit, starting from the highest position
  • The remainder of each step becomes part of the divisor of the next step.
  • Repeat this process until the entire calculation is complete

Application of Long Division

  • Integer division:Handling multi-digit division
  • Fractional division:Calculate the number of decimal places
  • Multinomial division:Division of Algebra
  • Euclidean division:Algorithmic Basis for Finding the Greatest Common Factor
Schematic diagram of the basic format of long division

Advantages of Long Division

  • Strong systematic: clear steps, easy to grasp
  • Wide applicability: can handle all types of divisions
  • High accuracy: Reduces the possibility of calculation errors.
  • Teaching value: Helps to understand the nature of division

Arithmetic long division - division of whole numbers and decimals

Long division of integers

Integer long division is the most basic form of long division and is used to handle multi-digit division operations.

Example: Calculate 1260257 ÷ 37

Example of calculation of long division of integers

Calculate the steps:

  1. Look at the first two digits of the divisor, 12, as 12
  2. 126 ÷ 37 = 3 remaining 15 (as 37 × 3 = 111, 126 - 111 = 15)
  3. Bring down the next digit 0 to get 150
  4. 150 ÷ 37 = 4 remaining 2 (as 37 × 4 = 148, 150 - 148 = 2)
  5. Continue this process until all the numbers have been processed.

Final answer: 1260257 ÷ 37 = 34061 餘 0

Long division of decimals

Long division with decimals is used to handle division operations with decimals in a similar way to long division with integers.

Example: Calculate 15.75 ÷ 2.5

Example of long division with decimals

Calculate the steps:

  1. Multiply the divisor 2.5 by 10 to make 25, and multiply the divisor 15.75 by 10 to make 157.5.
  2. Long division by whole numbers: 157.5 ÷ 25
  3. 25 × 6 = 150, 157.5 - 150 = 7.5
  4. 7.5 ÷ 25 = 0.3
  5. Final answer: 15.75 ÷ 2.5 = 6.3

Fractional long division technique: Multiply both the divisor and the divisor by the same number of times 10 to make the divisor a whole number, then divide by the whole number.

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Polynomial Long Division - Algebraic Arithmetic Skills

Polynomial long division is the method of dividing one polynomial by another, following the same principles as arithmetic long division. It is an important operation in algebra, especially when dealing with rational functions, factorisation and solving equations.

Standard Format and Schematic Steps for Polynomial Long Division

Basic Concepts of Polynomial Long Division

Basic formula:Divisor = Divisor × Quotient + Remainder

Important Notes:

  • Polynomials must be ordered in descending order (from highest to lowest).
  • Missing items must be filled in (e.g. x³ + 2x + 1 should be written as x³ + 0x² + 2x + 1)
  • The number of times the remainder of the formula must be less than the number of times the division of the formula is performed.
  • Eliminate the highest terms of the divisor at each step

Detailed example: (4x⁴ + 7x² - 3x + 5) ÷ (x² + 2x - 3)

Step 1: Set up the division format and add zeros

Divided by: 4x⁴ + 0x³ + 7x² - 3x + 5

Division: x² + 2x - 3

Step 2: First Division

4x⁴ ÷ x² = 4x²

Calculation: 4x² × (x² + 2x - 3) = 4x⁴ + 8x³ - 12x²

Subtraction: (4x⁴ + 0x³ + 7x²) - (4x⁴ + 8x³ - 12x²) = -8x³ + 19x²

Step 3: Second division

-8x³ ÷ x² = -8x

Calculation: -8x × (x² + 2x - 3) = -8x³ - 16x² + 24x

Subtract: (-8x³ + 19x² - 3x) - (-8x³ - 16x² + 24x) = 35x² - 27x

Step 4: Third division

35x² ÷ x² = 35

Calculation: 35 × (x² + 2x - 3) = 35x² + 70x - 105

Subtraction: (35x² - 27x + 5) - (35x² + 70x - 105) = -97x + 110

Example of a complete calculation of polynomial long division.

Step 5: Determine the final answer

Since the number of times (1) that the remainder -97x + 110 is less than the number of times (2) that the division x² + 2x - 3 is done, the division ends.

Final Answer:

Quotient: 4x² - 8x + 35

Remainder: -97x + 110

Verification: (x² + 2x - 3)(4x² - 8x + 35) + (-97x + 110) = 4x⁴ + 7x² - 3x + 5 ✓

Calculation Steps - Detailed Problem Solving Procedure

Preparation Stage

  • Arrange divisors and divisors in descending order
  • Completion of missing items (expressed as a coefficient of zero)
  • Setting the standard long division format

Calculation stage

  • Divide the first term of the divisor by the first term of the divisor.
  • Multiply the quotient by the whole divisor
  • Subtract the result of multiplication from the divisor.
  • Repeat the above steps until the number of remaining equations is less than the divisor.

Verification Stage

  • Check that the number of times the remainder of the formula is less than the divisor.
  • Formula: Divisor = Divisor × Quotient + Remainder
  • Substitution of calculation results for verification
  • Confirmation of the correctness of the calculation
Long Division Calculation Flowchart

Calculation Tips and Tricks

Symbol Handling

  • Note the change in plus and minus sign
  • Change all symbols when subtracting.
  • Maintaining Symbol Consistency

Principle of Alignment

  • Align the same items.
  • Maintaining a neat and tidy format
  • Easy to check and calculate

Checking Methods

  • Check immediately after each step
  • Use of validation formulas
  • Note the number of times the balance is limited

Pro Tip: When performing polynomial long division, using different coloured pens to mark the different steps can greatly reduce calculation errors.

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Common Errors - Avoiding Calculation Pitfalls

There are a number of typical errors that students make when performing long division calculations. Understanding these errors and learning to avoid them is the key to improving accuracy.

Formatting Error

Example of a long division format error

❌ Error: Forgot to fill in zeros

Write x³ + 2x + 1 directly in division format without adding 0x².

✅ Correct: full format

It should be written as x³ + 0x² + 2x + 1 to make sure that all the sub-items have corresponding positions.

Calculation Error

Example of a long division error

❌ Error: Symbol Handling Error

Forgetting to change symbols in a subtraction step, or inconsistent handling of symbols.

✅ Correct: careful handling of symbols

To subtract, change the sign of each item of the subtracted number and then add.

Conceptual error

Examples of Long Division Conceptual Errors

❌ Error: Residual count too high

Continues division even if the number of remainders is less than the number of divisors.

✅ Correct: Stop in time

When the number of times the remainder is less than the number of times the divisor is divided, the division ends.

Error Prevention Strategies

Systematic Inspection

  • Immediate check for each completed step
  • Confirmation of the correctness of symbols and counts
  • Maintaining a neat calculation format

Multi-Certification

  • Check your answer using the validation formula
  • Recalculating Critical Steps
  • Limit on the number of times the balance can be checked

Exercise Enhancement

  • Doing different types of exercises
  • Familiarise yourself with common calculation models
  • Cultivating Good Computing Habits

Remember: In polynomial long division, filling in missing zeros is the most overlooked step, but it is the key to avoiding computational errors.

DSE Applications - Exam Tips and Question Types

In the DSE Maths exam, long division mainly appears in the following modules:

  • More about Polynomials:Polynomial long division is the core element
  • Functions and images:Simplification of rational functions requires the use of polynomial division.
  • Number series and levels:Preparatory Knowledge of Partial Fractional Decomposition
  • Micro-accumulation:Preprocessing of rational function integrals
Examples of long division problems in the DSE Maths Exam

Exam Strategies and Tips

time management

  • Multinomial division usually takes 3-5 minutes.
  • Start with a quick estimate of the reasonableness of the answer
  • Allow time for validation
  • If the calculation is complex, consider other methods

Calculation Techniques

  • Keep the format neat and tidy for easy checking
  • Verify each step immediately after calculation
  • Note the correct handling of symbols
  • Simplifying Calculations by Factorisation

Answer Format

  • Clearly labelled quotient and residual
  • Write a complete verification process
  • Use standard mathematical symbols
  • Keep your answers simple and clear

Examples of DSE past questions

DSE Mathematics Paper 1, 2023

Title:Set f(x) = 2x³ - 5x² + 3x - 1 and g(x) = x - 2.

(a) Find the quotient and remainder equations for f(x) ÷ g(x).

(b) Using the result in (a), find the value of f(2).

Answer:

(a) Quotient: 2x² - x + 1, Remainder: 1

(b) f(2) = 1 (using the Remainder Theorem)

DSE Mathematics Paper 1, 2022

Title:It is known that the remainder of the polynomial P(x) divided by (x² - 1) is 3x + 2.

(a) Find the values of P(1) and P(-1).

(b) If P(x) = x³ + ax² + bx + c, find an expression for a, b and c.

Answer:

(a) P(1) = 5, P(-1) = -1

(b) a + b + c = 4

Strategies for Solving Long Division Problems in DSE Maths Exam

Scoring Points

Calculation Process (60%)

  • Correct division steps
  • Accurate Symbol Handling
  • Clear calculation format

Final Answer (30%)

  • Correct quotient and remainder equations
  • Appropriate forms of simplification
  • Reasonable Answer Form

Validation Process (10%)

  • Use of validation formulas
  • Checking the reasonableness of answers
  • Confirmation of calculation correctness

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Summary - The Key to Mastering Long Division

Long division is an important computational tool in mathematics. Whether you are dealing with division of whole numbers, division of decimals, or division of polynomials, you will follow similar principles and steps. Mastering long division not only improves your mathematical skills, but also provides a solid foundation for learning more advanced mathematical concepts.

Key Learning Points Review

  • Understand the basic principles and applications of long division
  • Knowledge of long division with integers, decimals and polynomials
  • Attention to common errors in the calculation process and ways to avoid them.
  • Familiar with long division problem types and problem solving strategies in the DSE Maths Exam
  • Improve calculation accuracy and speed through systematic practice.

Suggestions for Further Study

  • Do more long division problems of different types.
  • Combining long division with other mathematical concepts (e.g. factorisation, Remainder Theorem)
  • Try to use long division to solve real-world problems.
  • Regularly review long division calculation steps and techniques.
  • Seek guidance from professional tutors to solve learning problems.
Students' self-confidence after successfully mastering long division

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