%5E3-expanded.jpeg "(2x+4)^3 Expanded")
Expanding (2x + 4)³
Expanding a binomial raised to a power can be done using the Binomial Theorem or by repeated multiplication. Let's explore both methods to expand (2x + 4)³.
Method 1: Using the Binomial Theorem
The Binomial Theorem states:
(a + b)ⁿ = ∑(n choose k) a^(n-k) b^k, where (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!)
Applying this to (2x + 4)³:
- a = 2x
- b = 4
- n = 3
Therefore:
(2x + 4)³ = ∑(3 choose k) (2x)^(3-k) 4^k
Expanding the sum:
- k = 0: (3 choose 0) (2x)³ 4⁰ = 1 * 8x³ * 1 = 8x³
- k = 1: (3 choose 1) (2x)² 4¹ = 3 * 4x² * 4 = 48x²
- k = 2: (3 choose 2) (2x)¹ 4² = 3 * 2x * 16 = 96x
- k = 3: (3 choose 3) (2x)⁰ 4³ = 1 * 1 * 64 = 64
Adding all the terms together:
(2x + 4)³ = 8x³ + 48x² + 96x + 64
Method 2: Repeated Multiplication
We can also expand the expression by multiplying (2x + 4) by itself three times:
(2x + 4)³ = (2x + 4) * (2x + 4) * (2x + 4)
First, expand the first two factors:
(2x + 4) * (2x + 4) = 4x² + 8x + 8x + 16 = 4x² + 16x + 16
Now, multiply this result by (2x + 4):
(4x² + 16x + 16) * (2x + 4) = 8x³ + 32x² + 32x + 8x² + 64x + 64
Combining like terms:
(2x + 4)³ = 8x³ + 40x² + 96x + 64
Conclusion
Both methods lead to the same result: (2x + 4)³ = 8x³ + 40x² + 96x + 64. The Binomial Theorem offers a more structured and efficient approach, especially for higher powers. However, repeated multiplication can be helpful for understanding the process and visualizing the expansion.