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Give me note on long division. Stating the meaning of long divisions, the uses and much examples.
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Note on Long Division
Meaning
- Long division is a standard algorithm for dividing one number (the dividend) by another (the divisor) to find the quotient and remainder.
- It systematically breaks the division into a sequence of simpler steps: estimate a digit of the quotient, multiply, subtract, and bring down the next digit.
When to use / Uses
- Dividing large whole numbers by multi-digit divisors when a calculator is not wanted.
- Finding exact quotients and remainders.
- Converting a division that produces a repeating or terminating decimal.
- Converting division with decimal divisors into whole-number divisors (by shifting decimals).
- Dividing polynomials (polynomial long division) to find quotient and remainder or to factor polynomials.
- Checking answers by multiplication (divisor × quotient + remainder = dividend).
The algorithm / Steps (for whole numbers)
1. Write the dividend and the divisor in the long-division layout (divisor outside, dividend inside).
2. Starting from the leftmost digits of the dividend, find the smallest block of digits that the divisor can go into at least once.
3. Estimate how many times the divisor fits into that block (one digit of the quotient).
4. Multiply the divisor by that digit and write the product under the block; subtract to get a remainder.
5. Bring down the next digit of the dividend next to the remainder to form a new number.
6. Repeat steps 3–5 until all digits have been brought down.
7. If the remainder is nonzero but you need a decimal quotient, append a decimal point to the quotient, add zeros to the dividend (bring them down), and continue.
8. Remainder must always be smaller than the divisor (for integer quotients). Check: divisor × quotient + remainder = dividend.
Tips / common pitfalls
- Always put a zero in the quotient if the divisor does not go into a particular place value (place-holder zeros).
- Estimate quotient digits by round numbers or short division to avoid repeated trial multiplications.
- Keep the remainder smaller than the divisor. If not, increase the quotient digit.
- For decimal divisors, move the decimal point right in both divisor and dividend the same number of places to make the divisor a whole number, then divide.
Worked examples
1) 1548 ÷ 12
- 12 into 15 goes 1 → 1×12 = 12 → 15 − 12 = 3. Bring down 4 → 34.
- 12 into 34 goes 2 → 2×12 = 24 → 34 − 24 = 10. Bring down 8 → 108.
- 12 into 108 goes 9 → 9×12 = 108 → remainder 0.
- Quotient = 129. (Check: 12×129 = 1548)
2) 2005 ÷ 5
- 5 into 20 → 4 (remainder 0). Bring down 0 → 0 → quotient digit 0. Bring down 5 → 1.
- Quotient = 401.
3) 1003 ÷ 7
- 7 into 10 → 1, rem 3, bring down 0 → 30.
- 7 into 30 → 4, rem 2, bring down 3 → 23.
- 7 into 23 → 3, rem 2.
- Quotient = 143 remainder 2. So 1003 = 7×143 + 2.
4) 726 ÷ 4 (showing decimal)
- 4 into 7 → 1 rem 3, bring down 2 → 32 → 8 rem 0, bring down 6 → 6 → 1 rem 2.
- Integer quotient 181 remainder 2. If you want decimal: 181.5 (since 2/4 = 0.5).
5) 4321 ÷ 23 (multi-digit divisor)
- 23 into 43 → 1 rem 20, bring down 2 → 202.
- 23 into 202 → 8 (8×23 = 184) rem 18, bring down 1 → 181.
- 23 into 181 → 7 (7×23 = 161) rem 20.
- Quotient = 187 remainder 20. (Check: 23×187 + 20 = 4321)
6) Decimal divisor: 4.56 ÷ 0.12
- Move decimal 2 places right in both numbers → 456 ÷ 12.
- 12 into 45 → 3 rem 9, bring down 6 → 96 → 8.
- Quotient = 38.
7) Division producing repeating decimal: 37 ÷ 6
- 6 into 37 → 6 rem 1. For decimals: bring down 0 → 10 → 1 rem 4, bring down 0 → 40 → 6 rem 4 ... repeating.
- Decimal = 6.1666... = 6.1(6).
Polynomial long division (brief example)
Divide x^3 + 2x^2 − 5x + 6 by x − 1:
- x^3 ÷ x = x^2 → multiply (x^2)(x − 1) = x^3 − x^2. Subtract → remainder 3x^2 − 5x + 6.
- 3x^2 ÷ x = 3x → multiply → 3x^2 − 3x. Subtract → remainder −2x + 6.
- −2x ÷ x = −2 → multiply → −2x + 2. Subtract → remainder 4.
- Result: quotient x^2 + 3x − 2 with remainder 4. So (x^3 + 2x^2 −5x + 6) ÷ (x − 1) = x^2 + 3x − 2 + 4/(x − 1).
Practice problems (try these)
- 6789 ÷ 13
- 5040 ÷ 7
- 1234 ÷ 6
- 9.72 ÷ 0.03
- Divide x^3 − 4x + 4 by x − 2
Answers (brief)
- 6789 ÷ 13 = 522 remainder 3 (522 R3).
- 5040 ÷ 7 = 720.
- 1234 ÷ 6 = 205 remainder 4 (or 205.666...).
- 9.72 ÷ 0.03 = 324.
- (x^3 − 4x + 4) ÷ (x − 2) → quotient x^2 + 2x + 0 and remainder 0? Check quickly: divide x^3 by x = x^2; multiply x^2(x−2)=x^3 − 2x^2; subtract → 2x^2 − 4x; 2x^2 ÷ x = 2x; multiply → 2x^2 − 4x; subtract → 0 remainder 4? Wait: original has +4 constant; after subtract we bring down +4 so remainder 4. So quotient x^2 + 2x and remainder 4. (So result = x^2 + 2x + 4/(x − 2).)
If you want, I can:
- Show any of the numeric examples in the standard long-division layout step-by-step, or
- Give more practice problems with solutions, or
- Explain synthetic division as a shortcut for divisors of the form x − c.
Meaning
- Long division is a standard algorithm for dividing one number (the dividend) by another (the divisor) to find the quotient and remainder.
- It systematically breaks the division into a sequence of simpler steps: estimate a digit of the quotient, multiply, subtract, and bring down the next digit.
When to use / Uses
- Dividing large whole numbers by multi-digit divisors when a calculator is not wanted.
- Finding exact quotients and remainders.
- Converting a division that produces a repeating or terminating decimal.
- Converting division with decimal divisors into whole-number divisors (by shifting decimals).
- Dividing polynomials (polynomial long division) to find quotient and remainder or to factor polynomials.
- Checking answers by multiplication (divisor × quotient + remainder = dividend).
The algorithm / Steps (for whole numbers)
1. Write the dividend and the divisor in the long-division layout (divisor outside, dividend inside).
2. Starting from the leftmost digits of the dividend, find the smallest block of digits that the divisor can go into at least once.
3. Estimate how many times the divisor fits into that block (one digit of the quotient).
4. Multiply the divisor by that digit and write the product under the block; subtract to get a remainder.
5. Bring down the next digit of the dividend next to the remainder to form a new number.
6. Repeat steps 3–5 until all digits have been brought down.
7. If the remainder is nonzero but you need a decimal quotient, append a decimal point to the quotient, add zeros to the dividend (bring them down), and continue.
8. Remainder must always be smaller than the divisor (for integer quotients). Check: divisor × quotient + remainder = dividend.
Tips / common pitfalls
- Always put a zero in the quotient if the divisor does not go into a particular place value (place-holder zeros).
- Estimate quotient digits by round numbers or short division to avoid repeated trial multiplications.
- Keep the remainder smaller than the divisor. If not, increase the quotient digit.
- For decimal divisors, move the decimal point right in both divisor and dividend the same number of places to make the divisor a whole number, then divide.
Worked examples
1) 1548 ÷ 12
- 12 into 15 goes 1 → 1×12 = 12 → 15 − 12 = 3. Bring down 4 → 34.
- 12 into 34 goes 2 → 2×12 = 24 → 34 − 24 = 10. Bring down 8 → 108.
- 12 into 108 goes 9 → 9×12 = 108 → remainder 0.
- Quotient = 129. (Check: 12×129 = 1548)
2) 2005 ÷ 5
- 5 into 20 → 4 (remainder 0). Bring down 0 → 0 → quotient digit 0. Bring down 5 → 1.
- Quotient = 401.
3) 1003 ÷ 7
- 7 into 10 → 1, rem 3, bring down 0 → 30.
- 7 into 30 → 4, rem 2, bring down 3 → 23.
- 7 into 23 → 3, rem 2.
- Quotient = 143 remainder 2. So 1003 = 7×143 + 2.
4) 726 ÷ 4 (showing decimal)
- 4 into 7 → 1 rem 3, bring down 2 → 32 → 8 rem 0, bring down 6 → 6 → 1 rem 2.
- Integer quotient 181 remainder 2. If you want decimal: 181.5 (since 2/4 = 0.5).
5) 4321 ÷ 23 (multi-digit divisor)
- 23 into 43 → 1 rem 20, bring down 2 → 202.
- 23 into 202 → 8 (8×23 = 184) rem 18, bring down 1 → 181.
- 23 into 181 → 7 (7×23 = 161) rem 20.
- Quotient = 187 remainder 20. (Check: 23×187 + 20 = 4321)
6) Decimal divisor: 4.56 ÷ 0.12
- Move decimal 2 places right in both numbers → 456 ÷ 12.
- 12 into 45 → 3 rem 9, bring down 6 → 96 → 8.
- Quotient = 38.
7) Division producing repeating decimal: 37 ÷ 6
- 6 into 37 → 6 rem 1. For decimals: bring down 0 → 10 → 1 rem 4, bring down 0 → 40 → 6 rem 4 ... repeating.
- Decimal = 6.1666... = 6.1(6).
Polynomial long division (brief example)
Divide x^3 + 2x^2 − 5x + 6 by x − 1:
- x^3 ÷ x = x^2 → multiply (x^2)(x − 1) = x^3 − x^2. Subtract → remainder 3x^2 − 5x + 6.
- 3x^2 ÷ x = 3x → multiply → 3x^2 − 3x. Subtract → remainder −2x + 6.
- −2x ÷ x = −2 → multiply → −2x + 2. Subtract → remainder 4.
- Result: quotient x^2 + 3x − 2 with remainder 4. So (x^3 + 2x^2 −5x + 6) ÷ (x − 1) = x^2 + 3x − 2 + 4/(x − 1).
Practice problems (try these)
- 6789 ÷ 13
- 5040 ÷ 7
- 1234 ÷ 6
- 9.72 ÷ 0.03
- Divide x^3 − 4x + 4 by x − 2
Answers (brief)
- 6789 ÷ 13 = 522 remainder 3 (522 R3).
- 5040 ÷ 7 = 720.
- 1234 ÷ 6 = 205 remainder 4 (or 205.666...).
- 9.72 ÷ 0.03 = 324.
- (x^3 − 4x + 4) ÷ (x − 2) → quotient x^2 + 2x + 0 and remainder 0? Check quickly: divide x^3 by x = x^2; multiply x^2(x−2)=x^3 − 2x^2; subtract → 2x^2 − 4x; 2x^2 ÷ x = 2x; multiply → 2x^2 − 4x; subtract → 0 remainder 4? Wait: original has +4 constant; after subtract we bring down +4 so remainder 4. So quotient x^2 + 2x and remainder 4. (So result = x^2 + 2x + 4/(x − 2).)
If you want, I can:
- Show any of the numeric examples in the standard long-division layout step-by-step, or
- Give more practice problems with solutions, or
- Explain synthetic division as a shortcut for divisors of the form x − c.
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