%5E5.jpeg "(2x-5)^5")
Expanding (2x - 5)^5: A Journey Through the Binomial Theorem
The expression (2x - 5)^5 might seem daunting at first glance, but we can conquer it using the powerful Binomial Theorem. This theorem provides a systematic way to expand any binomial raised to a power.
Understanding the Binomial Theorem
The Binomial Theorem states:
(a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where:
- n is the power to which the binomial is raised
- k ranges from 0 to n
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
This formula might seem complex, but it has a simple pattern. Let's break it down:
- (n choose k) represents the number of ways to choose k elements from a set of n elements.
- a^(n-k) represents the first term of the binomial raised to the power (n-k).
- b^k represents the second term of the binomial raised to the power k.
Applying the Binomial Theorem to (2x - 5)^5
In our case, a = 2x, b = -5, and n = 5. Let's apply the theorem step by step:
- k = 0: (5 choose 0) * (2x)^5 * (-5)^0 = 1 * 32x^5 * 1 = 32x^5
- k = 1: (5 choose 1) * (2x)^4 * (-5)^1 = 5 * 16x^4 * -5 = -400x^4
- k = 2: (5 choose 2) * (2x)^3 * (-5)^2 = 10 * 8x^3 * 25 = 2000x^3
- k = 3: (5 choose 3) * (2x)^2 * (-5)^3 = 10 * 4x^2 * -125 = -5000x^2
- k = 4: (5 choose 4) * (2x)^1 * (-5)^4 = 5 * 2x * 625 = 6250x
- k = 5: (5 choose 5) * (2x)^0 * (-5)^5 = 1 * 1 * -3125 = -3125
Finally, we add all the terms together:
(2x - 5)^5 = 32x^5 - 400x^4 + 2000x^3 - 5000x^2 + 6250x - 3125
Conclusion
The Binomial Theorem is a powerful tool for expanding expressions like (2x - 5)^5. By understanding its components and applying the formula step by step, we can systematically arrive at the expanded form of the expression.