Binomial Theorem
The binomial theorem provides a way to expand expressions of the form \((a + b)^n\).
Binomial Theorem
For any positive integer \(n\):
\[ (a + b)^n = \sum_{k=0}^{n} C(n, k) a^{n-k} b^k \]
Where \(C(n, k) = \frac{n!}{k!(n-k)!}\) is the binomial coefficient.
Example
Expand \((x + 2)^3\):
\[ (x + 2)^3 = C(3, 0)x^3 2^0 + C(3, 1)x^2 2^1 + C(3, 2)x 2^2 + C(3, 3)x^0 2^3 \]
\[ = 1 \cdot x^3 + 3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 4 + 1 \cdot 8 = x^3 + 6x^2 + 12x + 8 \]
Applications
The binomial theorem is used in probability (e.g., binomial distribution), algebra, and computer graphics.