CAIE A-Level Mathematics 3.1.5 Binomial Expansion

CAIE A-Level Mathematics 3.1.5 Binomial Expansion
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise adapting (1 + u)ⁿ for rational powers, expanding in ascending powers, combining short series and deriving the strict interval for which each expansion is valid.

How this is tested

  • factor out the constant to rewrite the expression as A(1 + u)ⁿ before expanding
  • calculate explicit rational binomial coefficients and collect terms only to the required power
  • derive the strict validity interval by solving |u| < 1 for the original variable

Question 10(b)

[Maximum number: 5]

Let f(x)=x3+2x−11(3+x)(2+x2)\mathrm{f}(x)=\frac{x^{3}+2 x-11}{(3+x)\left(2+x^{2}\right)}.

Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x2x^{2}. [5]