Binomial Theorem

You already know how to expand (a + b)² = a² + 2ab + b². But what about (a + b)⁵ or (a + b)¹⁰? Multiplying out these expressions manually would be tedious. The Binomial Theorem gives us an elegant formula to expand any power of a binomial expression.

Pascal's Triangle

Pascal's triangle provides the coefficients for binomial expansions:

n=0:           1
n=1:          1  1
n=2:        1  2  1
n=3:       1  3  3  1
n=4:      1  4  6  4  1
n=5:    1  5  10  10  5  1

Each number is the sum of the two numbers directly above it.

The Binomial Theorem

For any positive integer n:

(a + b)ⁿ = C(n,0)aⁿ + C(n,1)aⁿ⁻¹b + C(n,2)aⁿ⁻²b² + ... + C(n,n)bⁿ

where C(n,r) = n! / [r!(n-r)!] is the binomial coefficient, also written as ⁿCr.

General Term

The (r+1)th term (general term) of (a + b)ⁿ is:

T(r+1) = C(n,r) aⁿ⁻ʳ bʳ

This formula lets you find any specific term without expanding the entire expression.

Worked Example 1: Full Expansion

Expand (x + 2)⁴.

Solution:

• (x + 2)⁴ = C(4,0)x⁴ + C(4,1)x³(2) + C(4,2)x²(2²) + C(4,3)x(2³) + C(4,4)(2⁴)
• = 1(x⁴) + 4(2x³) + 6(4x²) + 4(8x) + 1(16)
• = x⁴ + 8x³ + 24x² + 32x + 16

Worked Example 2: Finding a Specific Term

Find the 3rd term in the expansion of (2x - 3)⁵.

Solution:

• The 3rd term means r = 2 (since T(r+1))
• T3 = C(5,2)(2x)³(-3)²
• = 10 x 8x³ x 9
• = 720x³

Properties of Binomial Coefficients

• C(n,0) = C(n,n) = 1
• C(n,r) = C(n, n-r) (symmetry)
• Sum of all coefficients: C(n,0) + C(n,1) + ... + C(n,n) = 2ⁿ
• Number of terms in (a + b)ⁿ is (n + 1)

Middle Term

• If n is even: the middle term is T(n/2 + 1)
• If n is odd: there are two middle terms, T((n+1)/2) and T((n+3)/2)

Key Takeaways

• The binomial theorem expands (a + b)ⁿ using binomial coefficients
• General term: T(r+1) = C(n,r) aⁿ⁻ʳ bʳ
• Pascal's triangle gives binomial coefficients
• The expansion of (a + b)ⁿ has (n + 1) terms