Applications of Binomial Theorem

The binomial theorem makes it easy to find the coefficient of any term in \((a + b)^n\) without expanding the whole expression. Instead of writing all the terms, we directly use the general term formula:

\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]

The coefficient is simply the numerical part of the term.

The binomial theorem helps approximate values of expressions like \((1 + x)^n\) when \(x\) is small. This is useful because multiplying decimals repeatedly can be slow, but the first few terms of the expansion give a close estimate.

The idea is:

\[(1 + x)^n \approx 1 + nx + \dfrac{n(n-1)}{2}x^2 + \cdots \]

For small \(|x|\), higher powers like \(x^3, x^4\) become tiny and can be ignored for quick approximations.

Some algebraic expressions are easier to handle after expanding them with the binomial theorem. This method is especially useful when powers are large or when expressions contain combinations of variables.

Expansion helps reveal patterns or allows further simplification.

Sometimes we need the first few or last few terms of a binomial expansion, especially for approximations or expressions with very high powers.

The binomial theorem helps extract them quickly without expanding the entire series.