(x-y)^3 Binomial Expansion

Jasonbradley

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Jun 17, 2024

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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 non-negative 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^(n-k) 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! * (n-k)!)

Expanding (x - y)³

Let's apply the binomial theorem to expand (x - y)³.

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

2. Apply the formula:

   (x - y)³ = ∑_(k=0)³ (3 choose k) x^(3-k) (-y)^k

3. 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

4. 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

5. Substitute the values:

   (x - y)³ = 1 * x³ * 1 + 3 * x² * (-y) + 3 * x * y² + 1 * 1 * (-y)³

6. 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! * (n-k)!)
• 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.