The Binomial Theorem - Algebra-2

1. Fundamental Concepts

Definition: The Binomial Theorem is a formula that describes the algebraic expansion of powers of a binomial expression.
Binomial Coefficients: These are the coefficients in the expansion and can be found using combinations, denoted as $$\text{{C}}(n, k) = \frac{n!}{k!(n-k)!}$$.
Expansion Formula: $$(a + b)^n = \sum_{k=0}^{n} \text{{C}}(n, k) a^{n-k} b^k$$

Basic Rule: $$(a + b)^2 = a^2 + 2ab + b^2$$

Degree Preservation: The highest degree in the result matches input

Application: Used to expand expressions in various mathematical contexts, including probability and combinatorics

Problem: Expand $$(x + y)^3$$

Apply the Binomial Theorem: $$(x + y)^3 = \sum_{k=0}^{3} \text{{C}}(3, k) x^{3-k} y^k$$
Calculate each term:
- For $$k = 0$$: $$\text{{C}}(3, 0) x^3 y^0 = 1 \cdot x^3 \cdot 1 = x^3$$
- For $$k = 1$$: $$\text{{C}}(3, 1) x^2 y^1 = 3 \cdot x^2 \cdot y = 3x^2y$$
- For $$k = 2$$: $$\text{{C}}(3, 2) x^1 y^2 = 3 \cdot x \cdot y^2 = 3xy^2$$
- For $$k = 3$$: $$\text{{C}}(3, 3) x^0 y^3 = 1 \cdot 1 \cdot y^3 = y^3$$
Combine all terms: $$x^3 + 3x^2y + 3xy^2 + y^3$$

Validation: Substitute $$x = 1$$ and $$y = 1$$ → Original: $$(1 + 1)^3 = 8$$; Simplified: $$1 + 3 + 3 + 1 = 8$$ ✓

Problem: Expand $$(2a - b)^4$$

Apply the Binomial Theorem: $$(2a - b)^4 = \sum_{k=0}^{4} \text{{C}}(4, k) (2a)^{4-k} (-b)^k$$
Calculate each term:
- For $$k = 0$$: $$\text{{C}}(4, 0) (2a)^4 (-b)^0 = 1 \cdot 16a^4 \cdot 1 = 16a^4$$
- For $$k = 1$$: $$\text{{C}}(4, 1) (2a)^3 (-b)^1 = 4 \cdot 8a^3 \cdot (-b) = -32a^3b$$
- For $$k = 2$$: $$\text{{C}}(4, 2) (2a)^2 (-b)^2 = 6 \cdot 4a^2 \cdot b^2 = 24a^2b^2$$
- For $$k = 3$$: $$\text{{C}}(4, 3) (2a)^1 (-b)^3 = 4 \cdot 2a \cdot (-b^3) = -8ab^3$$
- For $$k = 4$$: $$\text{{C}}(4, 4) (2a)^0 (-b)^4 = 1 \cdot 1 \cdot b^4 = b^4$$
Combine all terms: $$16a^4 - 32a^3b + 24a^2b^2 - 8ab^3 + b^4$$

Validation: Substitute $$a = 1$$ and $$b = 1$$ → Original: $$(2 \cdot 1 - 1)^4 = 1$$; Simplified: $$16 - 32 + 24 - 8 + 1 = 1$$ ✓

Pattern Recognition: Identify patterns in binomial expansions to predict coefficients and terms.
Symmetric Properties: Use the symmetry of binomial coefficients to simplify calculations.
Substitution Method: Substitute simple values for variables to verify the correctness of expanded forms.