To prove division with a remainder, you demonstrate that the original number (dividend) can be expressed as the result of multiplying the divisor and quotient, plus the remainder. This method relies on the fundamental Division Algorithm.
Understanding the Division Algorithm
The Division Algorithm states that for any two integers, a (the dividend) and b (the divisor, where b is not zero), there exist unique integers q (the quotient) and r (the remainder) such that:
*a = b q + r**
where 0 ≤ r < |b| (The remainder 'r' is always non-negative and less than the absolute value of the divisor 'b').
This equation is at the heart of proving division with a remainder. It asserts that you can represent the dividend as a multiple of the divisor plus a remainder that's smaller than the divisor.
The Proof Process
The proof essentially involves two main steps:
- Find the quotient (q) and remainder (r): This is typically done through the standard long division process.
- Verify the relationship: Confirm that the equation *a = b q + r** holds true.
Practical Example
Let's divide 27 by 5:
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Division: When you divide 27 by 5, you get a quotient of 5 and a remainder of 2.
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Verification:
- a (dividend) = 27
- b (divisor) = 5
- q (quotient) = 5
- r (remainder) = 2
Now, check: 27 = 5 * 5 + 2. The equation holds true; 27 = 25 + 2.
Since 2 is less than 5, the condition of remainder is also met, which is: 0 ≤ r < |b|.
Proof with Polynomials
The same principle applies to polynomial division. According to the reference "Proof of Remainder Theorem", when a polynomial p(x) is divided by a linear polynomial (x-a), there will exist a quotient q(x) and a remainder r:
*p(x) = (x - a) q(x) + r**
The remainder, r, in this case, is equal to p(a).
- For example: If p(x) = x2 + 2x + 1 is divided by (x-1), then p(1) = 12 + 2(1) + 1 = 4. Thus, the remainder when p(x) is divided by x-1 is 4.
To prove this, you perform polynomial long division to find q(x) and r, then verify that the equation holds.
Key Concepts and Summary
| Concept | Description |
|---|---|
| Dividend (a) | The number being divided. |
| Divisor (b) | The number by which the dividend is divided. |
| Quotient (q) | The result of the division, ignoring any remainder. |
| Remainder (r) | The amount left over after division, always non-negative and less than the divisor. |
| Division Algorithm | The fundamental rule *a = b q + r**, where 0 ≤ r < |
Conclusion
Proving division with a remainder requires showing that the dividend can be accurately represented by the formula derived from the Division Algorithm. This equation highlights the relationship between the dividend, divisor, quotient, and remainder and holds true for both integers and polynomials. The remainder is always less than the divisor.
