Binomial Theorem/Examples/(a - 2 b)^4

Example of Use of Binomial Theorem

$\paren {a - 2 b}^4 = a^4 - 8 a^3 b + 24 a^2 b^2 - 32 a b^3 + 16 b^4$

$\ds \paren {a - 2 b}^4$

$=$

$\ds \sum_k \binom 4 k a^k \paren {2 b}^{4 - k}$

$\ds $

$=$

$\ds \binom 4 4 a^4 + \binom 4 3 a^3 \paren {2 b} + \binom 4 2 a^2 \paren {2 b}^2 + \binom 4 1 a \paren {2 b}^3 + \binom 4 0 \paren {2 b}^4$

$\ds $

$=$

$\ds a^4 + 4 a^3 \paren {2 b} + 6 a^2 \paren {2 b}^2 + 4 a \paren {2 b}^3 + \paren {2 b}^4$

$\ds $

$=$

$\ds \paren {a - 2 b}^4 = a^4 - 8 a^3 b + 24 a^2 b^2 - 32 a b^3 + 16 b^4$

$\blacksquare$

2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 3$: The Binomial Formula and Binomial Coefficients: Binomial Formula for Positive Integral $n$: Example