Theory question 111
Binomial Theorem(a) Find the binomial expansion of (1 + 2x)7 and (1 − 2x)7.
(b) Using the result in 11(a), find, correct to three decimal places, the value of (1.2)7 − (0.8)7.
Show the model answer
Model answer
(a) (1 + 2x)⁷ = 1 + 14x + 84x² + 280x³ + 560x⁴ + 672x⁵ + 448x⁶ + 128x⁷.
(1 − 2x)⁷ = 1 − 14x + 84x² − 280x³ + 560x⁴ − 672x⁵ + 448x⁶ − 128x⁷.
(b) Put x = 0.1: the difference = 2(14x + 280x³ + 672x⁵ + 128x⁷) = 2(1.4 + 0.28 + 0.00672 + 0.0000128) ≈ 3.373.
Explanation
Subtracting the two expansions cancels the even powers.