Pearson Edexcel IAL Mathematics P4.4.1 Binomial Series for any rational n

Practise expanding rational powers into binomial series, controlling coefficients, validity ranges and small-x approximations.

Exam points

rewrite (a+bx)^n as a(1+kx)^n before expanding in ascending powers of x
state the interval of validity from |kx|<1 for the rewritten expression
compare coefficients in a binomial series to find constants or an x^3 coefficient

[Maximum number: 4]

\mathrm{f}(x)=(8-3 x)^{\frac{4}{3}} \quad 0<x<\frac{8}{3}

Show that the binomial expansion of f(x) in ascending powers of x up to and including the term in $x^{3}$ is

A-8 x+\frac{x^{2}}{2}+B x^{3}+\ldots

where A and B are constants to be found.

$(8-3 x)^{\frac{4}{3}}=16\left(1-\frac{3}{8} x\right)^{\frac{4}{3}}$ but condone $16(1-k x)^{\frac{4}{3}}, k \neq 3$

M1

"Correct" term 3 or term 4 in
$(1-k x)^{\frac{4}{3}}=1 \pm \frac{4}{3} k x+\frac{\frac{4}{3} \times \frac{1}{3}}{2}( \pm k x)^{2}+\frac{\frac{4}{3} \times \frac{1}{3} \times\left(-\frac{2}{3}\right)}{6}( \pm k x)^{3}+\ldots$

M1

Two terms correct of $16-8 x+\frac{x^{2}}{2}+\frac{x^{3}}{24}+\ldots$ following M1, M1 and $k= \pm \frac{3}{8}$

A1

$=16-8 x+\frac{x^{2}}{2}+\frac{x^{3}}{24}+\ldots$

A1

(4)

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P4.4.1 - Binomial Series for any rational n question 1