O-Level Binomial expansions: worked solution

Question

Find, in ascending powers of x, the first four terms of the expansion of (a/x + 3x²)⁶, simplifying each term.

Use the binomial expansion (p + q)⁶ = Σ ⁶Cr p^(6-r) q^r for r = 0 to 6, taking p = a/x and q = 3x². Write down the general term and simplify the power of x before choosing which terms to keep: ⁶Cr (a/x)^(6-r) (3x²)^r = ⁶Cr a^(6-r) x^-(6-r) × 3^r x^(2r) = ⁶Cr a^(6-r) 3^r x^(r-6+2r) = ⁶Cr a^(6-r) 3^r x^(3r-6). So the power of x is 3r - 6, which INCREASES as r increases. Ascending powers of x therefore means taking r = 0, 1, 2, 3 in that order. r = 0: ⁶C0 a⁶ 3⁰ x^(-6) = 1 × a⁶ × 1 × x⁻⁶ = a⁶/x⁶. r = 1: ⁶C1 a⁵ 3¹ x^(-3) = 6 × a⁵ × 3 × x⁻³ = 18a⁵/x³. r = 2: ⁶C2 a⁴ 3² x^(0) = 15 × a⁴ × 9 × 1 = 135a⁴. r = 3: ⁶C3 a³ 3³ x^(3) = 20 × a³ × 27 × x³ = 540a³x³. (The binomial coefficients used are ⁶C0 = 1, ⁶C1 = 6, ⁶C2 = 15, ⁶C3 = 20.) So the first four terms in ascending powers of x are a⁶/x⁶ + 18a⁵/x³ + 135a⁴ + 540a³x³ + ...