%5E3-expand.jpeg "(2x-1)^3 Expand")
Expanding (2x-1)^3
The expression (2x-1)^3 represents the cube of the binomial (2x-1). To expand it, we can use the binomial theorem or simply apply the distributive property multiple times.
Using the Binomial Theorem
The binomial theorem states:
(a + b)^n = ∑ (n choose k) a^(n-k) b^k
where:
- n is a non-negative integer
- k is an integer from 0 to n
- (n choose k) is the binomial coefficient, calculated as n! / (k! (n-k)!)
Applying this to our expression:
(2x-1)^3 = ∑ (3 choose k) (2x)^(3-k) (-1)^k
Expanding the summation:
(3 choose 0) (2x)^3 (-1)^0 + (3 choose 1) (2x)^2 (-1)^1 + (3 choose 2) (2x)^1 (-1)^2 + (3 choose 3) (2x)^0 (-1)^3
Calculating the binomial coefficients and simplifying:
1 * 8x^3 * 1 + 3 * 4x^2 * -1 + 3 * 2x * 1 + 1 * 1 * -1
Therefore, the expanded form is:
(2x-1)^3 = 8x^3 - 12x^2 + 6x - 1
Expanding by Distributive Property
Alternatively, we can expand (2x-1)^3 by repeated application of the distributive property:
(2x-1)^3 = (2x-1)(2x-1)(2x-1)
First, expand the first two factors:
(2x-1)(2x-1) = 4x^2 - 2x - 2x + 1 = 4x^2 - 4x + 1
Then, multiply the result by (2x-1):
(4x^2 - 4x + 1)(2x-1) = 8x^3 - 8x^2 + 2x - 8x^2 + 8x - 1
Combining like terms:
(2x-1)^3 = 8x^3 - 12x^2 + 6x - 1
Conclusion
Both methods lead to the same result, (2x-1)^3 = 8x^3 - 12x^2 + 6x - 1. The binomial theorem provides a more systematic and concise approach, while the distributive property offers a more intuitive step-by-step expansion.