Expanding (x+4)^6: A Journey Through Binomial Theorem
Expanding expressions like (x+4)^6 can seem daunting, but with the help of the Binomial Theorem, it becomes a manageable task. Let's explore how to break down this expression stepbystep.
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^(nk) * b^k
Where:
 (n choose k) represents the binomial coefficient, calculated as n! / (k! * (nk)!).
 k ranges from 0 to n.
Applying the Theorem to (x+4)^6

Identify a and b: In our case, a = x and b = 4.

Determine n: The exponent is 6, so n = 6.

Calculate binomial coefficients: We need to calculate (6 choose k) for k = 0, 1, 2, 3, 4, 5, and 6.
 (6 choose 0) = 6! / (0! * 6!) = 1
 (6 choose 1) = 6! / (1! * 5!) = 6
 (6 choose 2) = 6! / (2! * 4!) = 15
 (6 choose 3) = 6! / (3! * 3!) = 20
 (6 choose 4) = 6! / (4! * 2!) = 15
 (6 choose 5) = 6! / (5! * 1!) = 6
 (6 choose 6) = 6! / (6! * 0!) = 1

Apply the formula: Now, we can plug in the values into the Binomial Theorem formula:
(x + 4)^6 = (6 choose 0) * x^6 * 4^0 + (6 choose 1) * x^5 * 4^1 + (6 choose 2) * x^4 * 4^2 + (6 choose 3) * x^3 * 4^3 + (6 choose 4) * x^2 * 4^4 + (6 choose 5) * x^1 * 4^5 + (6 choose 6) * x^0 * 4^6
 Simplify:
(x + 4)^6 = x^6 + 24x^5 + 240x^4 + 1280x^3 + 3840x^2 + 6144x + 4096
Conclusion
By applying the Binomial Theorem, we have successfully expanded (x + 4)^6. Remember, the key is understanding the formula and following the steps carefully. While the expansion may seem complex at first, it's a powerful tool for working with polynomials and their powers.