O-Level Binomial expansions: worked solution

Question

Taking n = 7, demonstrate that the product (48/x² − x⁴)(√x − 2/x)⁷ contains no constant term.

A constant term is the term in x⁰. Rather than expanding all eight terms of (√x − 2/x)⁷, write down its general term and pick out only the powers that are needed. By the binomial theorem, for r = 0, 1, 2, ..., 7 the general term of (√x − 2/x)⁷ is C(7, r) × (√x)⁷⁻ʳ × (−2/x)ʳ Simplify the powers of x in each factor separately. (√x)⁷⁻ʳ = (x^(1/2))⁷⁻ʳ = x^((7−r)/2) (−2/x)ʳ = (−2)ʳ × x⁻ʳ Multiplying, the powers of x add: general term = C(7, r)(−2)ʳ x^((7−r)/2 − r) = C(7, r)(−2)ʳ x^((7−3r)/2) The first bracket has only two terms, 48x⁻² and −x⁴, so a constant in the product can arise in only two ways. Way 1: 48x⁻² must multiply a term in x², because x⁻² × x² = x⁰. Set the power equal to 2: (7 − 3r)/2 = 2 7 − 3r = 4 3r = 3, so r = 1 Its coefficient is C(7, 1)(−2)¹ = 7 × (−2) = −14, so the x² term is −14x². Way 2: −x⁴ must multiply a term in x⁻⁴, because x⁴ × x⁻⁴ = x⁰. Set the power equal to −4: (7 − 3r)/2 = −4 7 − 3r = −8 3r = 15, so r = 5 Its coefficient is C(7, 5)(−2)⁵ = 21 × (−32) = −672, so the x⁻⁴ term is −672x⁻⁴. Now multiply out the two contributions and add them. 48x⁻² × (−14x²) = 48 × (−14) = −672 (−x⁴) × (−672x⁻⁴) = (−1) × (−672) = +672 constant term = −672 + 672 = 0 The coefficient of x⁰ is zero, so the product (48/x² − x⁴)(√x − 2/x)⁷ has no constant term, as required.