(x+2y)^6

Jasonbradley

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Jun 16, 2024

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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^(n-k) * y^k

where:

• n is a non-negative integer (the power)
• k is an integer ranging from 0 to n
• (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)

Applying the Theorem to (x + 2y)^6

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

2. 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

3. 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

4. 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.