(x-2y)^4 Binomial Expansion

Jasonbradley

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Jun 17, 2024

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Binomial Expansion of (x-2y)^4

The binomial theorem provides a formula to expand expressions of the form $(x + y)^n$, where n is a non-negative integer. Let's explore how to apply this theorem to expand $(x-2y)^4$.

Understanding the Binomial Theorem

The binomial theorem states:

(x + y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k

where:

• (n choose k) is the binomial coefficient, calculated as n!/(k!*(n-k)!), representing the number of ways to choose k items from a set of n items.
• ∑_(k=0)^n means we sum the terms from k=0 to k=n.

Applying the Theorem to (x-2y)^4

1. Identify n: In our case, n = 4.

2. Expand the summation: We need to calculate the terms from k=0 to k=4.

3. Calculate the binomial coefficients:

   • (4 choose 0) = 4!/(0!4!) = 1
   • (4 choose 1) = 4!/(1!3!) = 4
   • (4 choose 2) = 4!/(2!2!) = 6
   • (4 choose 3) = 4!/(3!1!) = 4
   • (4 choose 4) = 4!/(4!0!) = 1

4. Substitute into the formula:

(x - 2y)^4 = (4 choose 0) * x^4 * (-2y)^0 + (4 choose 1) * x^3 * (-2y)^1 + (4 choose 2) * x^2 * (-2y)^2 + (4 choose 3) * x^1 * (-2y)^3 + (4 choose 4) * x^0 * (-2y)^4

5. Simplify:

(x - 2y)^4 = 1 * x^4 * 1 + 4 * x^3 * (-2y) + 6 * x^2 * 4y^2 + 4 * x * (-8y^3) + 1 * 1 * 16y^4

(x - 2y)^4 = x^4 - 8x^3y + 24x^2y^2 - 32xy^3 + 16y^4

Conclusion

Therefore, the binomial expansion of (x-2y)^4 is x^4 - 8x^3y + 24x^2y^2 - 32xy^3 + 16y^4. This process highlights the power of the binomial theorem in expanding complex expressions with ease.