%5E5.jpeg "(2x^3+1)^5")
Expanding (2x³ + 1)⁵: A Journey into Binomial Theorem
The expression (2x³ + 1)⁵ presents a challenge when trying to expand it directly. Multiplying it out five times would be incredibly tedious. Instead, we can utilize the Binomial Theorem to make this process significantly easier.
Understanding the Binomial Theorem
The Binomial Theorem provides a formula for expanding expressions of the form (x + y)ⁿ. It 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)!).
- ∑_(k=0)^n signifies the summation from k=0 to k=n.
Applying the Theorem to (2x³ + 1)⁵
Let's break down how to use the Binomial Theorem to expand (2x³ + 1)⁵.
- Identify x and y: In our case, x = 2x³ and y = 1.
- Determine n: The power is 5, so n = 5.
Now, we can apply the theorem and expand the expression:
(2x³ + 1)⁵ = ∑_(k=0)^5 (5 choose k) (2x³)^(5-k) (1)^k
Let's expand this summation:
- k = 0: (5 choose 0) (2x³)^(5-0) (1)^0 = 1 * 32x¹⁵ * 1 = 32x¹⁵
- k = 1: (5 choose 1) (2x³)^(5-1) (1)^1 = 5 * 16x¹² * 1 = 80x¹²
- k = 2: (5 choose 2) (2x³)^(5-2) (1)² = 10 * 8x⁹ * 1 = 80x⁹
- k = 3: (5 choose 3) (2x³)^(5-3) (1)³ = 10 * 4x⁶ * 1 = 40x⁶
- k = 4: (5 choose 4) (2x³)^(5-4) (1)⁴ = 5 * 2x³ * 1 = 10x³
- k = 5: (5 choose 5) (2x³)^(5-5) (1)⁵ = 1 * 1 * 1 = 1
Finally, combining all the terms, the fully expanded form of (2x³ + 1)⁵ is:
(2x³ + 1)⁵ = 32x¹⁵ + 80x¹² + 80x⁹ + 40x⁶ + 10x³ + 1
Conclusion
By leveraging the Binomial Theorem, we successfully expanded (2x³ + 1)⁵. This method is highly efficient and eliminates the need for extensive multiplication. Understanding the theorem and its application is crucial for simplifying similar expressions in algebra and calculus.