%5E4-expanded.jpeg "(2x-1)^4 Expanded")
Expanding (2x - 1)^4
Expanding expressions like (2x - 1)^4 can be done using the Binomial Theorem or by 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 = ∑ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n and (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).
Applying this to our expression (2x - 1)^4, we have:
(2x - 1)^4 = (4 choose 0) * (2x)^4 * (-1)^0 + (4 choose 1) * (2x)^3 * (-1)^1 + (4 choose 2) * (2x)^2 * (-1)^2 + (4 choose 3) * (2x)^1 * (-1)^3 + (4 choose 4) * (2x)^0 * (-1)^4
Calculating the binomial coefficients and simplifying:
(2x - 1)^4 = 1 * 16x^4 * 1 + 4 * 8x^3 * -1 + 6 * 4x^2 * 1 + 4 * 2x * -1 + 1 * 1 * 1
Therefore, (2x - 1)^4 = 16x^4 - 32x^3 + 24x^2 - 8x + 1
Expanding by Repeated Multiplication
We can also expand (2x - 1)^4 by multiplying it out repeatedly:
(2x - 1)^4 = (2x - 1) * (2x - 1) * (2x - 1) * (2x - 1)
First, multiply the first two factors:
(2x - 1) * (2x - 1) = 4x^2 - 4x + 1
Then, multiply this result by the third factor:
(4x^2 - 4x + 1) * (2x - 1) = 8x^3 - 12x^2 + 6x - 1
Finally, multiply this result by the last factor:
(8x^3 - 12x^2 + 6x - 1) * (2x - 1) = 16x^4 - 32x^3 + 24x^2 - 8x + 1
As you can see, both methods lead to the same result: (2x - 1)^4 = 16x^4 - 32x^3 + 24x^2 - 8x + 1.
Conclusion
The Binomial Theorem provides a more systematic and efficient way to expand expressions like (2x - 1)^4, especially for higher powers. However, understanding the concept of repeated multiplication is essential for grasping the foundation of the Binomial Theorem.