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Give me full notes on long division. Please make the note nice. Please
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Long Division — Complete, easy-to-follow notes
OVERVIEW
- Long division is a step-by-step algorithm for dividing numbers (or polynomials) to find a quotient and possibly a remainder.
- It’s useful for dividing large numbers by smaller numbers, converting remainders to decimals, and dividing polynomials.
KEY TERMS
- Dividend: the number being divided.
- Divisor: the number you divide by.
- Quotient: the result of division.
- Remainder: what’s left after division when the divisor doesn’t go evenly into the dividend.
STANDARD LONG-DIVISION ALGORITHM (whole numbers)
1. Write the divisor at the left and place the dividend under the division bracket (or inside the parenthesis).
2. Look at the leftmost digits of the dividend that make a number equal to or larger than the divisor.
3. Estimate how many times the divisor fits into that number (this is one digit of the quotient).
4. Multiply the divisor by that digit and write the product under the number you used.
5. Subtract to get a remainder for that step.
6. Bring down the next digit of the dividend and repeat steps 3–5 until all digits have been brought down.
7. The digits placed on top form the quotient. If the last subtraction leaves a nonzero value and there are no more digits to bring down, that value is the remainder.
CHECK: Multiply the quotient by the divisor and add the remainder. That should equal the original dividend.
Example 1 — Simple, single-digit divisor
Divide 123 by 4.
Setup:
4 ) 123
Steps:
- 4 into 1? 0 times -> consider first two digits 12.
- 4 into 12? 3 times. Put 3 in the quotient.
- 3 × 4 = 12. Subtract 12 from 12 → 0. Bring down 3.
- 4 into 3? 0 times. Put 0 in quotient. Subtract 0. Remainder is 3.
Result: quotient = 30, remainder = 3. (Check: 30×4 + 3 = 120 + 3 = 123)
Example 2 — Multi-digit divisor
Divide 789 by 23.
Steps:
- 23 into 78? 3 times (3×23=69). Write 3.
- Subtract 69 → remainder 9. Bring down 9 → 99.
- 23 into 99? 4 times (4×23=92). Write 4.
- Subtract 92 → remainder 7. No more digits to bring down.
Result: quotient = 34, remainder = 7. (Check: 34×23 + 7 = 782 + 7 = 789)
REMAINDERS: express remainder as
- “quotient R remainder” (e.g., 34 R 7),
- a fraction remainder/divisor (34 7/23), or
- a decimal by continuing division (see decimals).
DIVISION WITH DECIMALS
- When the dividend runs out but you still have a remainder, place a decimal point in the quotient directly above the decimal point in the dividend (or insert a decimal point in the quotient when you start bringing down zeros).
- Append zeros to the dividend as needed and continue.
Example — 7 ÷ 4
- 4 into 7 → 1 (1×4=4), remainder 3. Put decimal point and bring down 0 → 30.
- 4 into 30 → 7 (7×4=28), remainder 2. Bring down 0 → 20.
- 4 into 20 → 5 (5×4=20), remainder 0.
Result: 1.75
CONVERTING REMAINDER TO DECIMAL
- Continue the long division by bringing down zeros; each zero gives another decimal place.
CONVERTING REMAINDER TO FRACTION
- remainder / divisor, then simplify. Example: 123 ÷ 4 = 30 remainder 3 = 30 3/4 = 30.75.
SPECIAL CASES & RULES
- Divisor = 0: Division undefined (cannot divide by zero).
- Divisor > Dividend: Quotient = 0 and remainder = dividend (for integer division) or you can give a decimal quotient < 1.
- Negative numbers: Divide absolute values, then apply sign rules (quotient negative if exactly one of divisor/dividend negative; positive if both negative or both positive).
ESTIMATION AND TRIALS
- For multi-digit divisors, estimate how many times the divisor fits into the chunk by comparing leading digits (or use mental multiplication).
- If the product is too large, reduce the trial digit.
ALTERNATIVE: PARTIAL QUOTIENTS METHOD (good for learning/mental math)
- Subtract large chunks of divisor multiples from the dividend and keep track of how much you subtracted. Add those chunk counts to get the quotient.
- Example: 789 ÷ 23:
- Subtract 23×10=230 → 789-230=559 (quotient part: 10)
- Subtract 23×10=230 → 559-230=329 (quotient part: 10 more)
- Subtract 23×10=230 → 329-230=99 (quotient part: 10 more)
- Subtract 23×4=92 → 99-92=7 (quotient part: 4)
- Total quotient 10+10+10+4 = 34, remainder 7.
COMMON MISTAKES
- Not aligning digits of quotient with dividend digits.
- Forgetting to bring down the next digit.
- Placing the decimal point incorrectly in the quotient.
- Multiplying by the wrong quotient digit during a step.
- Not checking the final answer.
SHORT TIPS
- Always start with the smallest leftmost group of digits at least as large as the divisor.
- If stuck, try one less in the trial digit.
- Keep computations neat — long division relies on correct alignment.
- Check with multiplication at the end.
POLYNOMIAL LONG DIVISION (brief — similar idea)
- Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
- Multiply the entire divisor by that quotient term, subtract from the dividend, and bring down the next term.
- Repeat until the degree of the remainder is less than the degree of the divisor.
Example — divide x^3 − 2x^2 + 0x + 4 by x − 3:
- x^3 / x = x^2 → multiply: (x−3)×x^2 = x^3 − 3x^2. Subtract → remainder x^2 + 0x.
- x^2 / x = x → multiply: (x−3)×x = x^2 − 3x. Subtract → remainder 3x + 4.
- 3x / x = 3 → multiply: (x−3)×3 = 3x − 9. Subtract → remainder 13.
Result: quotient = x^2 + x + 3, remainder = 13. So (x^3 − 2x^2 + 4) ÷ (x − 3) = x^2 + x + 3 + 13/(x − 3).
SYNTHETIC DIVISION
- A shortcut for dividing by linear terms of the form x − c. Faster and uses only coefficients.
PRACTICE PROBLEMS (try them, then check answers below)
Integer arithmetic
1) 487 ÷ 5
2) 1024 ÷ 8
3) 3456 ÷ 12
4) 2001 ÷ 37
5) 957 ÷ 13
Decimals & remainders
6) 5 ÷ 6 (give decimal to 3 places)
7) 250 ÷ 7 (give quotient and remainder)
8) 12.5 ÷ 2.5
Polynomial
9) Divide 2x^3 + 3x^2 − x + 5 by x + 2
10) Divide x^4 − x^3 + 0x^2 + 4 by x − 1
ANSWERS
1) 97 remainder 2 (97 R 2) → check: 97×5 + 2 = 487
2) 128
3) 288
4) 54 remainder 3 (54 R 3) → 54×37 + 3 = 1998 + 3 = 2001
5) 73 remainder 8 (73 R 8) → 73×13 + 8 = 949 + 8 = 957
6) 0.833 (0.833 to 3 dp; true value 0.833333…)
7) 35 remainder 5 (35 R 5)
8) 5
9) Perform polynomial division: quotient = 2x^2 - x + 1, remainder = 7. (So 2x^3+3x^2−x+5 = (x+2)(2x^2−x+1) + 7)
10) Quotient = x^3 + 0x^2 - x + 1 (i.e., x^3 - x + 1), remainder = 5. (Check via division)
If you’d like, I can:
- Show step-by-step long division for any example you pick,
- Provide printable worksheets,
- Teach synthetic division with worked examples,
- Make the notes into a nicely formatted PDF or cheat-sheet. Which would you prefer?
OVERVIEW
- Long division is a step-by-step algorithm for dividing numbers (or polynomials) to find a quotient and possibly a remainder.
- It’s useful for dividing large numbers by smaller numbers, converting remainders to decimals, and dividing polynomials.
KEY TERMS
- Dividend: the number being divided.
- Divisor: the number you divide by.
- Quotient: the result of division.
- Remainder: what’s left after division when the divisor doesn’t go evenly into the dividend.
STANDARD LONG-DIVISION ALGORITHM (whole numbers)
1. Write the divisor at the left and place the dividend under the division bracket (or inside the parenthesis).
2. Look at the leftmost digits of the dividend that make a number equal to or larger than the divisor.
3. Estimate how many times the divisor fits into that number (this is one digit of the quotient).
4. Multiply the divisor by that digit and write the product under the number you used.
5. Subtract to get a remainder for that step.
6. Bring down the next digit of the dividend and repeat steps 3–5 until all digits have been brought down.
7. The digits placed on top form the quotient. If the last subtraction leaves a nonzero value and there are no more digits to bring down, that value is the remainder.
CHECK: Multiply the quotient by the divisor and add the remainder. That should equal the original dividend.
Example 1 — Simple, single-digit divisor
Divide 123 by 4.
Setup:
4 ) 123
Steps:
- 4 into 1? 0 times -> consider first two digits 12.
- 4 into 12? 3 times. Put 3 in the quotient.
- 3 × 4 = 12. Subtract 12 from 12 → 0. Bring down 3.
- 4 into 3? 0 times. Put 0 in quotient. Subtract 0. Remainder is 3.
Result: quotient = 30, remainder = 3. (Check: 30×4 + 3 = 120 + 3 = 123)
Example 2 — Multi-digit divisor
Divide 789 by 23.
Steps:
- 23 into 78? 3 times (3×23=69). Write 3.
- Subtract 69 → remainder 9. Bring down 9 → 99.
- 23 into 99? 4 times (4×23=92). Write 4.
- Subtract 92 → remainder 7. No more digits to bring down.
Result: quotient = 34, remainder = 7. (Check: 34×23 + 7 = 782 + 7 = 789)
REMAINDERS: express remainder as
- “quotient R remainder” (e.g., 34 R 7),
- a fraction remainder/divisor (34 7/23), or
- a decimal by continuing division (see decimals).
DIVISION WITH DECIMALS
- When the dividend runs out but you still have a remainder, place a decimal point in the quotient directly above the decimal point in the dividend (or insert a decimal point in the quotient when you start bringing down zeros).
- Append zeros to the dividend as needed and continue.
Example — 7 ÷ 4
- 4 into 7 → 1 (1×4=4), remainder 3. Put decimal point and bring down 0 → 30.
- 4 into 30 → 7 (7×4=28), remainder 2. Bring down 0 → 20.
- 4 into 20 → 5 (5×4=20), remainder 0.
Result: 1.75
CONVERTING REMAINDER TO DECIMAL
- Continue the long division by bringing down zeros; each zero gives another decimal place.
CONVERTING REMAINDER TO FRACTION
- remainder / divisor, then simplify. Example: 123 ÷ 4 = 30 remainder 3 = 30 3/4 = 30.75.
SPECIAL CASES & RULES
- Divisor = 0: Division undefined (cannot divide by zero).
- Divisor > Dividend: Quotient = 0 and remainder = dividend (for integer division) or you can give a decimal quotient < 1.
- Negative numbers: Divide absolute values, then apply sign rules (quotient negative if exactly one of divisor/dividend negative; positive if both negative or both positive).
ESTIMATION AND TRIALS
- For multi-digit divisors, estimate how many times the divisor fits into the chunk by comparing leading digits (or use mental multiplication).
- If the product is too large, reduce the trial digit.
ALTERNATIVE: PARTIAL QUOTIENTS METHOD (good for learning/mental math)
- Subtract large chunks of divisor multiples from the dividend and keep track of how much you subtracted. Add those chunk counts to get the quotient.
- Example: 789 ÷ 23:
- Subtract 23×10=230 → 789-230=559 (quotient part: 10)
- Subtract 23×10=230 → 559-230=329 (quotient part: 10 more)
- Subtract 23×10=230 → 329-230=99 (quotient part: 10 more)
- Subtract 23×4=92 → 99-92=7 (quotient part: 4)
- Total quotient 10+10+10+4 = 34, remainder 7.
COMMON MISTAKES
- Not aligning digits of quotient with dividend digits.
- Forgetting to bring down the next digit.
- Placing the decimal point incorrectly in the quotient.
- Multiplying by the wrong quotient digit during a step.
- Not checking the final answer.
SHORT TIPS
- Always start with the smallest leftmost group of digits at least as large as the divisor.
- If stuck, try one less in the trial digit.
- Keep computations neat — long division relies on correct alignment.
- Check with multiplication at the end.
POLYNOMIAL LONG DIVISION (brief — similar idea)
- Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
- Multiply the entire divisor by that quotient term, subtract from the dividend, and bring down the next term.
- Repeat until the degree of the remainder is less than the degree of the divisor.
Example — divide x^3 − 2x^2 + 0x + 4 by x − 3:
- x^3 / x = x^2 → multiply: (x−3)×x^2 = x^3 − 3x^2. Subtract → remainder x^2 + 0x.
- x^2 / x = x → multiply: (x−3)×x = x^2 − 3x. Subtract → remainder 3x + 4.
- 3x / x = 3 → multiply: (x−3)×3 = 3x − 9. Subtract → remainder 13.
Result: quotient = x^2 + x + 3, remainder = 13. So (x^3 − 2x^2 + 4) ÷ (x − 3) = x^2 + x + 3 + 13/(x − 3).
SYNTHETIC DIVISION
- A shortcut for dividing by linear terms of the form x − c. Faster and uses only coefficients.
PRACTICE PROBLEMS (try them, then check answers below)
Integer arithmetic
1) 487 ÷ 5
2) 1024 ÷ 8
3) 3456 ÷ 12
4) 2001 ÷ 37
5) 957 ÷ 13
Decimals & remainders
6) 5 ÷ 6 (give decimal to 3 places)
7) 250 ÷ 7 (give quotient and remainder)
8) 12.5 ÷ 2.5
Polynomial
9) Divide 2x^3 + 3x^2 − x + 5 by x + 2
10) Divide x^4 − x^3 + 0x^2 + 4 by x − 1
ANSWERS
1) 97 remainder 2 (97 R 2) → check: 97×5 + 2 = 487
2) 128
3) 288
4) 54 remainder 3 (54 R 3) → 54×37 + 3 = 1998 + 3 = 2001
5) 73 remainder 8 (73 R 8) → 73×13 + 8 = 949 + 8 = 957
6) 0.833 (0.833 to 3 dp; true value 0.833333…)
7) 35 remainder 5 (35 R 5)
8) 5
9) Perform polynomial division: quotient = 2x^2 - x + 1, remainder = 7. (So 2x^3+3x^2−x+5 = (x+2)(2x^2−x+1) + 7)
10) Quotient = x^3 + 0x^2 - x + 1 (i.e., x^3 - x + 1), remainder = 5. (Check via division)
If you’d like, I can:
- Show step-by-step long division for any example you pick,
- Provide printable worksheets,
- Teach synthetic division with worked examples,
- Make the notes into a nicely formatted PDF or cheat-sheet. Which would you prefer?
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