(x-y)^3 Expand Binomial

Jasonbradley

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Jun 17, 2024

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Expanding the Binomial (x - y)^3

The binomial theorem provides a formula for expanding expressions of the form (x + y)^n, where n is a non-negative integer. Let's explore how to apply this to expand (x - y)^3.

Understanding 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 representing the power of the binomial.
• k is an integer ranging from 0 to n.
• (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).

Applying the Binomial Theorem to (x - y)^3

1. Identify n: In our case, n = 3.

2. Expand the summation: We'll expand the summation for k = 0, 1, 2, and 3.

3. Calculate the binomial coefficients:

   • (3 choose 0) = 3! / (0! * 3!) = 1
   • (3 choose 1) = 3! / (1! * 2!) = 3
   • (3 choose 2) = 3! / (2! * 1!) = 3
   • (3 choose 3) = 3! / (3! * 0!) = 1

4. Substitute values and simplify: (x - y)^3 = (3 choose 0) * x^(3-0) * (-y)^0 + (3 choose 1) * x^(3-1) * (-y)^1 + (3 choose 2) * x^(3-2) * (-y)^2 + (3 choose 3) * x^(3-3) * (-y)^3 = 1 * x^3 * 1 + 3 * x^2 * (-y) + 3 * x * y^2 + 1 * 1 * (-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3

Conclusion

By applying the binomial theorem, we have successfully expanded (x - y)^3 into x^3 - 3x^2y + 3xy^2 - y^3. This method can be generalized to expand any binomial raised to a non-negative integer power.