Expanding Expressions Using the Binomial THeorem

Understanding the Binomial Theorem

The Binomial Theorem, also known as Newton’s Binomial Theorem, allows for the expansion of binomials raised to any power.
A binomial is an algebraic expression containing two terms, usually written in the format (a + b).
The Binomial Theorem states that (a + b)ⁿ is equal to the sum of [n choose k]·aⁿ⁻ᵏ·bᵏ for k = 0,1,…,n where [n choose k], also known as binomial coefficients, are the entries of Pascal’s triangle.

Binomial Coefficients & Pascal’s Triangle

Binomial coefficients are numbers that tell you how many ways there are of choosing k items from n options.
These numbers create a pattern called Pascal’s Triangle where each number is the sum of the two numbers directly above it.
A row ‘n’ in Pascal’s Triangle contains coefficients for the expansion of (a + b)ⁿ.

Expanding Expressions Using the Binomial Theorem

Expanding binomial expressions involves replacing ‘a’, ‘b’ and ‘n’ in the Binomial Theorem by the terms of the binomial and the power to which it is raised.
The expanded form of a binomial expression is a sequence of terms where the exponents of ‘a’ decrease and the exponents of ‘b’ increase from term to term.
Each term of the expansion multiplies the corresponding binomial coefficient from row ‘n’ of Pascal’s Triangle.
This expansion simplifies the evaluation of expressions and aids in carrying out operations like addition and multiplication on the binomial expression.

Example

Expand (2x - 3)³ using the Binomial Theorem: For n=3, the coefficients from Pascal’s Triangle are 1, 3, 3, 1. So, the expansion is 1·(2x)³·(-3)⁰ + 3·(2x)²·(-3)¹ + 3·(2x)¹·(-3)² + 1·(2x)⁰·(-3)³. This simplifies to 8x³ - 36x² + 54x - 27.

Note: Mastery of the Binomial Theorem requires a strong understanding of basic algebra, including working with exponent rules and polynomial expressions, as well as a familiarity with combinatorial setups like Pascal’s Triangle.