%5E5.jpeg "(2a-1)^5")
Expanding (2a - 1)^5: A Detailed Explanation
Expanding expressions like (2a - 1)^5 can be tedious using direct multiplication. However, the Binomial Theorem provides a powerful tool for this task. Let's break down the process step-by-step:
Understanding the Binomial Theorem
The Binomial Theorem states that for any real numbers x and y, and any non-negative integer n:
(x + y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k
Where (n choose k) represents the binomial coefficient, calculated as:
(n choose k) = n! / (k! * (n-k)!)
Applying the Theorem to (2a - 1)^5
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Identify x and y: In our case, x = 2a and y = -1.
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Determine n: The exponent is 5, so n = 5.
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Expand the summation:
- For k = 0: (5 choose 0) * (2a)^5 * (-1)^0 = 32a^5
- For k = 1: (5 choose 1) * (2a)^4 * (-1)^1 = -80a^4
- For k = 2: (5 choose 2) * (2a)^3 * (-1)^2 = 80a^3
- For k = 3: (5 choose 3) * (2a)^2 * (-1)^3 = -40a^2
- For k = 4: (5 choose 4) * (2a)^1 * (-1)^4 = 10a
- For k = 5: (5 choose 5) * (2a)^0 * (-1)^5 = -1
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Combine the terms: (2a - 1)^5 = 32a^5 - 80a^4 + 80a^3 - 40a^2 + 10a - 1
Conclusion
Using the Binomial Theorem, we successfully expanded (2a - 1)^5. This method avoids lengthy multiplications and provides a structured approach to handling such expressions.