%5E6-binomial-theorem.jpeg "(x+2y)^6 Binomial Theorem")
Expanding (x + 2y)⁶ with the Binomial Theorem
The binomial theorem provides a powerful tool for expanding expressions of the form (x + y)ⁿ. Let's explore how it works when expanding (x + 2y)⁶.
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) is the binomial coefficient, calculated as n!/(k!(n-k)!). It represents the number of ways to choose k objects from a set of n objects.
- ∑_(k=0)^n represents the sum from k=0 to n.
Expanding (x + 2y)⁶
Let's apply the binomial theorem to our expression:
(x + 2y)⁶ = ∑_(k=0)^6 (6 choose k) x^(6-k) (2y)^k
We need to calculate each term in the sum:
k = 0: (6 choose 0) x⁶ (2y)⁰ = 1 * x⁶ * 1 = x⁶
k = 1: (6 choose 1) x⁵ (2y)¹ = 6 * x⁵ * 2y = 12x⁵y
k = 2: (6 choose 2) x⁴ (2y)² = 15 * x⁴ * 4y² = 60x⁴y²
k = 3: (6 choose 3) x³ (2y)³ = 20 * x³ * 8y³ = 160x³y³
k = 4: (6 choose 4) x² (2y)⁴ = 15 * x² * 16y⁴ = 240x²y⁴
k = 5: (6 choose 5) x¹ (2y)⁵ = 6 * x¹ * 32y⁵ = 192xy⁵
k = 6: (6 choose 6) x⁰ (2y)⁶ = 1 * 1 * 64y⁶ = 64y⁶
Therefore, the complete expansion of (x + 2y)⁶ is:
(x + 2y)⁶ = x⁶ + 12x⁵y + 60x⁴y² + 160x³y³ + 240x²y⁴ + 192xy⁵ + 64y⁶
Conclusion
The binomial theorem provides a systematic way to expand expressions of the form (x + y)ⁿ. By understanding the formula and applying it step-by-step, we can efficiently obtain the expanded form of any binomial expression.