Beyond FOIL
How do you expand \((x+y)^{10}\)? Multiplying it out by hand is impossible. The Binomial Theorem allows us to find any term in the expansion using combinations (\(n \text{Cr}\)).
\[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]
Pascal's Triangle
The coefficients of the expansion follow a beautiful geometric pattern where each number is the sum of the two above it.
- \(n=0\): 1
- \(n=1\): 1, 1
- \(n=2\): 1, 2, 1 (This is \(x^2 + 2xy + y^2\))
- \(n=3\): 1, 3, 3, 1
Worked Examples
Example 1: Expanding a Cube
Expand \((x + 2)^3\).
- Coefficients from row 3: 1, 3, 3, 1.
- Powers of x: \(x^3, x^2, x^1, x^0\).
- Powers of 2: \(2^0, 2^1, 2^2, 2^3\).
- Combined: \(1(x^3)(1) + 3(x^2)(2) + 3(x)(4) + 1(1)(8)\).
- Result: \(x^3 + 6x^2 + 12x + 8\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often deal with "Small Perturbations"—tiny changes to a system. To calculate the effect of these changes, we use the Binomial Theorem to expand terms like \((1 + \epsilon)^n\) where \(\epsilon\) is very small. By keeping only the first two terms (\(1 + n\epsilon\)), we can approximate the behavior of complex atoms. This is the origin of Perturbation Theory, the method used to calculate almost every real-world value in quantum physics, from the magnetic moment of the electron to the energy of a black hole.