%5E4.jpeg "(2k+1)^4")
Expanding (2k + 1)^4
Expanding the expression (2k + 1)^4 can be achieved through various methods, such as the binomial theorem or repeated multiplication. Let's explore both methods:
1. Using the Binomial Theorem
The binomial theorem states that for any real numbers x and y, and any non-negative integer n:
(x + y)^n = ∑(n choose k) * x^(n-k) * y^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 (2k + 1)^4:
(2k + 1)^4 = ∑(4 choose k) * (2k)^(4-k) * 1^k
Let's calculate each term:
- k = 0: (4 choose 0) * (2k)^4 * 1^0 = 1 * 16k^4 * 1 = 16k^4
- k = 1: (4 choose 1) * (2k)^3 * 1^1 = 4 * 8k^3 * 1 = 32k^3
- k = 2: (4 choose 2) * (2k)^2 * 1^2 = 6 * 4k^2 * 1 = 24k^2
- k = 3: (4 choose 3) * (2k)^1 * 1^3 = 4 * 2k * 1 = 8k
- k = 4: (4 choose 4) * (2k)^0 * 1^4 = 1 * 1 * 1 = 1
Therefore, the expanded form of (2k + 1)^4 is:
(2k + 1)^4 = 16k^4 + 32k^3 + 24k^2 + 8k + 1
2. Repeated Multiplication
We can also expand (2k + 1)^4 by multiplying the expression by itself four times:
(2k + 1)^4 = (2k + 1) * (2k + 1) * (2k + 1) * (2k + 1)
Let's multiply the first two terms:
(2k + 1) * (2k + 1) = 4k^2 + 4k + 1
Now, let's multiply this result by the third term:
(4k^2 + 4k + 1) * (2k + 1) = 8k^3 + 12k^2 + 6k + 1
Finally, multiply by the last term:
(8k^3 + 12k^2 + 6k + 1) * (2k + 1) = 16k^4 + 32k^3 + 24k^2 + 8k + 1
We arrive at the same result as obtained using the binomial theorem.
Conclusion
Expanding (2k + 1)^4 can be achieved through the binomial theorem or repeated multiplication. Both methods lead to the same result: 16k^4 + 32k^3 + 24k^2 + 8k + 1. This expanded form is valuable for various applications in algebra and calculus.