%5E4.jpeg "(4x-1)^4")
Expanding (4x-1)^4
Expanding the expression (4x-1)^4 involves applying the binomial theorem or using repeated multiplication. Let's explore both methods:
Using the Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form (a + b)^n:
(a + b)^n = Σ (n choose k) * a^(n-k) * b^k
Where:
- (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).
- Σ signifies the sum from k = 0 to n.
Applying this to (4x - 1)^4:
- Identify a and b: a = 4x, b = -1
- Determine n: n = 4
Expanding the expression:
(4x - 1)^4 = (4 choose 0) * (4x)^4 * (-1)^0 + (4 choose 1) * (4x)^3 * (-1)^1 + (4 choose 2) * (4x)^2 * (-1)^2 + (4 choose 3) * (4x)^1 * (-1)^3 + (4 choose 4) * (4x)^0 * (-1)^4
Simplifying the binomial coefficients and exponents:
= 1 * 256x^4 * 1 + 4 * 64x^3 * -1 + 6 * 16x^2 * 1 + 4 * 4x * -1 + 1 * 1 * 1
= 256x^4 - 256x^3 + 96x^2 - 16x + 1
Expanding by Repeated Multiplication
Alternatively, we can expand (4x-1)^4 through repeated multiplication:
- (4x-1)^2 = (4x-1)(4x-1) = 16x^2 - 8x + 1
- (4x-1)^3 = (4x-1)^2 * (4x-1) = (16x^2 - 8x + 1)(4x-1) = 64x^3 - 48x^2 + 12x - 1
- (4x-1)^4 = (4x-1)^3 * (4x-1) = (64x^3 - 48x^2 + 12x - 1)(4x-1) = 256x^4 - 256x^3 + 96x^2 - 16x + 1
This method demonstrates the distributive property, but it can be more tedious for higher powers.
Conclusion
Both methods lead to the same result: (4x - 1)^4 = 256x^4 - 256x^3 + 96x^2 - 16x + 1. The binomial theorem provides a more efficient and systematic way for expanding binomial expressions with higher powers.