Expand (a + b)ⁿ efficiently using the binomial theorem.
The Binomial Theorem provides a formula for expanding (a + b)ⁿ without repeated multiplication. The coefficients are the binomial coefficients C(n, k), which also form Pascal's Triangle.
Pascal's Triangle (first 5 rows)
Each entry is the sum of the two entries above it. Row n gives coefficients for (a+b)ⁿ.
(x+2)⁴ = C(4,0)x⁴(2)⁰ + C(4,1)x³(2)¹ + C(4,2)x²(2)² + C(4,3)x¹(2)³ + C(4,4)x⁰(2)⁴ = x⁴ + 8x³ + 24x² + 32x + 16.
Remember This!
To find a specific term in (a+b)ⁿ without full expansion: the (k+1)th term is C(n,k)·aⁿ⁻ᵏ·bᵏ. This is useful when n is large.
What is the coefficient of x² in the expansion of (x + 3)⁵?