CAIE A-Level Mathematics 3.1.5 Binomial Expansion

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.

Exam points

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

[Maximum number: 5]

Let $\mathrm{f}(x)=\frac{2+11 x-10 x^{2}}{(1+2 x)(1-2 x)(2+x)}$ .

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

Use a correct method to find the first two terms of the expansion of one of
$(1+2x)^{-1},\quad (1-2x)^{-1},\quad (2+x)^{-1},\quad \left(1+\frac12x\right)^{-1}.$
M1

Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction.
A1FT + A1FT + A1FT

The follow-through is on A, B, and C.

Final answer:
$1+5x-\frac72x^2.$
A1

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3.1.5—Binomial expansion question 1