Understanding the Factor (a - b) in (a^n - b^n): A Deep Dive into Polynomial Division and Factor Theorems
Have you ever wondered how one first recognized that (a - b) is a factor of (a^n - b^n)? This article delves into the historical and mathematical reasoning behind this fascinating concept. We will explore the proof using the Factor Theorem and polynomial long division, providing a comprehensive understanding of why (a - b) is indeed a factor of (a^n - b^n).
1. Introduction to Polynomial Division and Factor Theorem
Polynomial division and the Factor Theorem are powerful tools in algebra that help us understand the factors of polynomials. The Factor Theorem states that if a polynomial f(x) has a root at x r, then (x - r) is a factor of f(x).
2. Proof Using the Factor Theorem
Factor Theorem: The Factor Theorem states that if f(x) is a polynomial and r is a root of f(x) then (x - r) is a factor of f(x).
Setting Up the Polynomial: Consider the polynomial f(x) x^n - b^n. We can rewrite this as f(a) a^n - b^n.
Evaluating at b: If we substitute a b into f(x), we get:
[f(b) b^n - b^n 0]
Conclusion from the Factor Theorem: Since f(b) 0, it follows that (a - b) is a factor of a^n - b^n.
3. Alternative Proof Using Polynomial Long Division
Another way to derive the same result is by using explicit polynomial division. We can write:
[a^n - b^n (a - b)(a^{n-1} a^{n-2}b a^{n-3}b^2 ldots ab^{n-2} b^{n-1})]
Verification: You can verify this by multiplying the right-hand side:
[(a - b)(a^{n-1} a^{n-2}b a^{n-3}b^2 ldots ab^{n-2} b^{n-1}) a^n - b^n]
Conclusion: This confirms that (a - b) is indeed a factor of a^n - b^n.
4. Historical and Intuitive Origin of the Concept
Reflecting on how one might come up with this, the process could be gradual. By first noting that the statement is true for n 2, a simpler case to verify, and then realizing it holds for n 3, one might be led to generalize for any n.
Another approach might be to start with x^n - 1 and generalize. By considering x - 1^n x^n - ldots - 1^n, one might think that there is a way to reverse-engineer a factorization that hits x - 1 and cancels out the middle terms. This typically involves including terms like x^{n-1} and 1 initially to get:
[(x - 1)x^{n-1}1 x^n - x^{n-1} - x 1]
However, to cancel the x^{n-1} term, one might want to include x^{n-2}, leading to:
[(x - 1)x^{n-1}x^{n-2} ldots 1]
Through trial and error, one might find that including additional terms will eventually lead to the desired factorization:
[(x - 1)(x^{n-1} x^{n-2} ldots x 1) x^n - 1]
This insight might then be generalized to:
[(a - b)(a^{n-1} a^{n-2}b a^{n-3}b^2 ldots ab^{n-2} b^{n-1}) a^n - b^n]
Which is the factorization we sought.
5. Conclusion
In summary, (a - b) is a factor of (a^n - b^n) because substituting a b results in zero, confirming that (a - b) divides (a^n - b^n). This can also be shown explicitly by factoring (a^n - b^n) into the product of (a - b) and another polynomial. This process reflects both historical reasoning and modern mathematical analysis, providing a robust understanding of this fundamental algebraic concept.