%5E3-binomial-expansion.jpeg "(2x+3)^3 Binomial Expansion")
Expanding (2x+3)³ using the Binomial Theorem
The binomial theorem provides a systematic way to expand expressions of the form (x + y)ⁿ for any positive integer n. Let's explore how to apply it to expand (2x + 3)³.
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)!). This formula essentially describes the expansion of (x + y)ⁿ as a sum of terms, each with a specific coefficient and powers of x and y.
Expanding (2x + 3)³
- Identify n: In our case, n = 3.
- Apply the formula: We need to calculate the terms for k = 0, 1, 2, and 3.
Let's break down each term:
- k = 0: (3 choose 0) (2x)³ (3)⁰ = 1 * 8x³ * 1 = 8x³
- k = 1: (3 choose 1) (2x)² (3)¹ = 3 * 4x² * 3 = 36x²
- k = 2: (3 choose 2) (2x)¹ (3)² = 3 * 2x * 9 = 54x
- k = 3: (3 choose 3) (2x)⁰ (3)³ = 1 * 1 * 27 = 27
- Combine the terms: Adding all the terms together, we get:
(2x + 3)³ = 8x³ + 36x² + 54x + 27
Conclusion
By applying the binomial theorem, we successfully expanded (2x + 3)³ to obtain the polynomial 8x³ + 36x² + 54x + 27. This method provides a structured and efficient approach to expanding any binomial expression raised to a positive integer power.