(1%2Ba)(1%2Ba%5E2)(1%2Ba%5E4).jpeg "(1-a)(1+a)(1+a^2)(1+a^4)")
Unveiling the Pattern: Expanding (1-a)(1+a)(1+a^2)(1+a^4)
This expression might look intimidating at first, but we can simplify it using a clever pattern and a bit of algebra. Let's break it down step by step:
Recognizing the Difference of Squares
The first two terms, (1-a) and (1+a), are a classic example of the difference of squares pattern. Remember, this pattern states:
(a - b)(a + b) = a² - b²
Applying this to our expression:
(1-a)(1+a) = 1² - a² = 1 - a²
Expanding Further
Now our expression becomes:
(1 - a²)(1 + a²)(1 + a⁴)
Notice another difference of squares pattern between (1 - a²) and (1 + a²):
(1 - a²)(1 + a²) = 1² - (a²)² = 1 - a⁴
Final Simplification
Finally, we have:
(1 - a⁴)(1 + a⁴) = 1² - (a⁴)² = 1 - a⁸
Therefore, the simplified form of (1-a)(1+a)(1+a^2)(1+a^4) is 1 - a⁸.
Key Takeaways
This problem demonstrates the power of recognizing patterns in algebra. By applying the difference of squares pattern multiple times, we were able to simplify the expression significantly. This technique can be applied to various algebraic expressions, making them easier to manipulate and solve.