%5E5-binomial-expansion.jpeg "(2x-3)^5 Binomial Expansion")
Expanding the Binomial: (2x - 3)^5
The binomial theorem provides a powerful tool for expanding expressions of the form (x + y)^n, where n is a positive integer. Let's apply this to expand the binomial (2x - 3)^5.
Understanding the Binomial Theorem
The binomial theorem states:
(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! / (k! * (n-k)!). It indicates the number of ways to choose k items from a set of n items.
Applying the Binomial Theorem
Let's break down the expansion of (2x - 3)^5 step by step:
-
Identify x and y: In our case, x = 2x and y = -3.
-
Expand using the formula: We apply the binomial theorem with n = 5:
(2x - 3)^5 = (5 choose 0) (2x)^5 (-3)^0 + (5 choose 1) (2x)^4 (-3)^1 + (5 choose 2) (2x)^3 (-3)^2 + (5 choose 3) (2x)^2 (-3)^3 + (5 choose 4) (2x)^1 (-3)^4 + (5 choose 5) (2x)^0 (-3)^5
- Calculate the binomial coefficients:
- (5 choose 0) = 1
- (5 choose 1) = 5
- (5 choose 2) = 10
- (5 choose 3) = 10
- (5 choose 4) = 5
- (5 choose 5) = 1
- Simplify the terms:
(2x - 3)^5 = 1 * 32x^5 * 1 + 5 * 16x^4 * -3 + 10 * 8x^3 * 9 + 10 * 4x^2 * -27 + 5 * 2x * 81 + 1 * 1 * -243
- Combine the terms:
**(2x - 3)^5 = ** 32x^5 - 240x^4 + 720x^3 - 1080x^2 + 810x - 243
Therefore, the expanded form of (2x - 3)^5 is 32x^5 - 240x^4 + 720x^3 - 1080x^2 + 810x - 243.
Conclusion
The binomial theorem provides a structured approach to expand expressions involving binomials raised to a power. By understanding the formula and applying it step by step, we can effectively expand complex expressions like (2x - 3)^5. This process is essential in various areas of mathematics, including algebra, calculus, and probability.