Expanding (x + 2y)^6 using the Binomial Theorem
The binomial theorem provides a powerful formula to expand expressions of the form (x + y)^n. Let's apply it to (x + 2y)^6.
The Binomial Theorem
The binomial theorem states:
(x + y)^n = ∑(n choose k) * x^(nk) * y^k
where:
 n is a nonnegative integer (the power)
 k is an integer ranging from 0 to n
 (n choose k) is the binomial coefficient, calculated as n! / (k! * (nk)!)
Applying the Theorem to (x + 2y)^6

Identify n and y: In our case, n = 6 and y = 2y.

Calculate the binomial coefficients:
 (6 choose 0) = 1
 (6 choose 1) = 6
 (6 choose 2) = 15
 (6 choose 3) = 20
 (6 choose 4) = 15
 (6 choose 5) = 6
 (6 choose 6) = 1

Apply the formula:
(x + 2y)^6 = (6 choose 0) * x^6 * (2y)^0 + (6 choose 1) * x^5 * (2y)^1 + (6 choose 2) * x^4 * (2y)^2 + (6 choose 3) * x^3 * (2y)^3 + (6 choose 4) * x^2 * (2y)^4 + (6 choose 5) * x^1 * (2y)^5 + (6 choose 6) * x^0 * (2y)^6

Simplify:
(x + 2y)^6 = x^6 + 12x^5y + 60x^4y^2 + 160x^3y^3 + 240x^2y^4 + 192xy^5 + 64y^6
Conclusion
By utilizing the binomial theorem, we successfully expanded (x + 2y)^6 to obtain the polynomial: x^6 + 12x^5y + 60x^4y^2 + 160x^3y^3 + 240x^2y^4 + 192xy^5 + 64y^6. This method provides a systematic and efficient way to expand binomial expressions raised to any power.