%5E3.jpeg "(4x-1)^3")
Expanding (4x - 1)³
This article will explore the expansion of the expression (4x - 1)³.
Understanding the Problem
We need to expand the expression (4x - 1)³, which means we need to multiply the binomial (4x - 1) by itself three times.
The Expansion
We can use the binomial theorem or the distributive property to expand this expression.
Using the Binomial Theorem
The binomial theorem states that for any positive integer n:
(x + y)ⁿ = ∑(k=0)^n (n choose k) x^(n-k) y^k
Where (n choose k) represents the binomial coefficient, which is calculated as n!/(k!(n-k)!).
In our case, x = 4x, y = -1, and n = 3. Applying the binomial theorem:
(4x - 1)³ = ∑(k=0)^3 (3 choose k) (4x)^(3-k) (-1)^k
Expanding this sum:
- k = 0: (3 choose 0) (4x)³ (-1)⁰ = 64x³
- k = 1: (3 choose 1) (4x)² (-1)¹ = -48x²
- k = 2: (3 choose 2) (4x)¹ (-1)² = 12x
- k = 3: (3 choose 3) (4x)⁰ (-1)³ = -1
Therefore, the expansion is:
(4x - 1)³ = 64x³ - 48x² + 12x - 1
Using the Distributive Property
We can also expand the expression by applying the distributive property multiple times:
(4x - 1)³ = (4x - 1) * (4x - 1) * (4x - 1)
First, we multiply the first two terms:
(4x - 1) * (4x - 1) = 16x² - 8x + 1
Then, we multiply this result by (4x - 1):
(16x² - 8x + 1) * (4x - 1) = 64x³ - 48x² + 12x - 1
As you can see, we get the same result using either method.
Conclusion
The expanded form of (4x - 1)³ is 64x³ - 48x² + 12x - 1.