Binomial Expansion Guide
What is a Binomial Expansion?
A binomial expansion is the process of expanding an expression of the form (a + b)^n using the binomial theorem.
The Binomial Theorem
The binomial theorem states that:
(a + b)^n = Σ(k=0 to n) (n choose k) a^(n-k) b^k
Where (n choose k) is the binomial coefficient, calculated as n!/(k!(n-k)!).
Example Calculation
For (x + 2)^3:
- (3 choose 0) x^3 × 2^0 = 1 × x^3 × 1 = x^3
- (3 choose 1) x^2 × 2^1 = 3 × x^2 × 2 = 6x^2
- (3 choose 2) x^1 × 2^2 = 3 × x × 4 = 12x
- (3 choose 3) x^0 × 2^3 = 1 × 1 × 8 = 8
Therefore, (x + 2)^3 = x^3 + 6x^2 + 12x + 8
Pascal's Triangle
Pascal's Triangle helps find binomial coefficients easily:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
Each number is the sum of the two numbers above it. The nth row gives the coefficients for (a+b)^n.