Chapter 28: The Binomial Theorem

The Binomial Theorem provides a general formula for the expansion of (a + b)ⁿ, where n is any positive integer. This theorem is a cornerstone of algebra, forming the basis for much of modern probability and calculus.

28.1 The Patterns of Expansion

By observing the expansion of small powers, we notice several consistent patterns:

The number of terms in the expansion is n + 1.
The exponent of a begins at n and decreases to 0.
The exponent of b begins at 0 and increases to n.
The sum of the exponents in each term is always n.

28.2 The General Formula

The expansion is written as follows:

(a + b)ⁿ = aⁿ + naⁿ⁻¹b + [n(n-1)/1·2]aⁿ⁻²b² + [n(n-1)(n-2)/1·2·3]aⁿ⁻³b³ + ...

28.3 Pascal's Triangle

The coefficients of the terms (the numbers in front of the variables) can also be found using Pascal's Triangle. Each number in the triangle is the sum of the two numbers directly above it.

        1

        1  1

        1  2  1

        1  3  3  1

        1  4  6  4  1

Example: Expand (x + 2)³.

Using coefficients 1, 3, 3, 1 from the triangle:
1(x)³ + 3(x)²(2)¹ + 3(x)¹(2)² + 1(2)³

Result: x³ + 6x² + 12x + 8

Doctor's Note: The Binomial Theorem also applies when the sign is negative. For (a - b)ⁿ, simply alternate the signs of the terms: +, -, +, -, ...

Doctor's Pro-Tip: Pay close attention to the coefficients. Because the triangle is symmetrical, the coefficients of the first and last terms are always 1, the second and second-to-last are always n, and so on. This "mirror effect" is a great way to verify your expansion!