The Binomial Theorem

Today we learned about the binomial theorem. This theorem allows you to expand x+y into a sum using this formula.


picture from wikipedia



REMEMBER! A binomial expansion where the exponent is n will have a n + 1 terms when expanded.

Example:

(x² + 3)⁶ = ₆C₀ (x²)⁶(3)⁰ + ₆C₁ (x²)⁵(3)¹ + ₆C₂ (x²)⁴(3)² + ₆C₃ (x²)³(3)³

+ ₆C₄ (x²)²(3)⁴ + ₆C₅ (x²)¹(3)⁵ + ₆C₆ (x²)⁰(3)⁶

Then simplifying gives me

(1)(x¹²)(1) + (6)(x¹⁰)(3) + (15)(x⁸)(9) + (20)(x⁶)(27)

+ (15)(x⁴)(81) + (6)(x²)(243) + (1)(1)(729)

= x¹² + 18x¹⁰ + 135x⁸ + 540x⁶ + 1215x⁴ + 1458x² + 729

When looking for a specific term within a binomial expansion we use this formula:

Example: Find the 14th term of (x+y)^16

a = x, b = y, n = 16, k = 14-1 = 13

t 13 + 1 = 16C13 x^(16-13) y^13

= 560 x^3 y^13