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Binomial Expansion

  • A Bionomial Expansion is a linear polynomial raised to a power, like this (a + b)n. As n increases, a pattern emerges in the coefficients of each term.
http://www.math10.com/en/algebra/probabilities/binomial-theorem/imgFig1.gif
  • The coefficients form a pattern called Pascal's Triangle, where each number is the sum of the two numbers above it.
http://alevelnotes.com/content_images/Pascal's_triangle_5.svg
  • For example, (3 + x)3 can be expanded to
    1 × 33 + 3 × 32x1 + 3 × 31x2 + 1 × 30x3
    = 27 + 27x + 9x2 + x3
  • There is another what of calculating the coefficients of each term, using Factorials and the Choose Function. The Choose function is written as xCy, and is calculated by
    where '!' donates a factorial function (e.g. 5! = 1×2×3×4×5).
  • To calculate the coefficient of the zth term of a binomial with exponent n, you would calculate nC(z-1).
    For example, the coefficient of the third term in the expansion (a + b)11 is 11C2 = 55
    Many calculators have a Choose Function programmed in them. It might look like nCr.