Understanding the Binomial Expansion of (x  y)³
The binomial theorem provides a systematic way to expand expressions of the form (x + y)ⁿ, where n is a nonnegative integer. In this case, we'll focus on expanding (x  y)³, which is a special case of the binomial theorem.
The Binomial Theorem
The binomial theorem states that:
(x + y)ⁿ = ∑_(k=0)^n (n choose k) x^(nk) y^k
where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items. It's calculated as:
(n choose k) = n! / (k! * (nk)!)
Expanding (x  y)³
Let's apply the binomial theorem to expand (x  y)³.

Identify n: In this case, n = 3.

Apply the formula:
(x  y)³ = ∑_(k=0)³ (3 choose k) x^(3k) (y)^k

Expand the summation:
(x  y)³ = (3 choose 0) x³ (y)⁰ + (3 choose 1) x² (y)¹ + (3 choose 2) x¹ (y)² + (3 choose 3) x⁰ (y)³

Calculate 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

Substitute the values:
(x  y)³ = 1 * x³ * 1 + 3 * x² * (y) + 3 * x * y² + 1 * 1 * (y)³

Simplify:
(x  y)³ = x³  3x²y + 3xy²  y³
Key Points
 The binomial theorem provides a systematic way to expand expressions with multiple terms raised to a power.
 The binomial coefficients can be calculated using the formula (n choose k) = n! / (k! * (nk)!)
 When expanding (x  y)³, remember to include the negative sign in front of y and its powers.
By understanding the binomial theorem and its application to (x  y)³, you can expand similar expressions and gain valuable insights into their algebraic structure.