Pearson Edexcel IAL Mathematics P4.4 Binomial expansion Question Bank

Practise binomial expansions with rational powers, simplified coefficients, validity ranges and approximations formed from combined series.

Exam points

state the validity range from |kx|<1 after rewriting the binomial expression
use a series to show approximations or find unknown coefficients in combined expressions

Covered syllabus points

P4.4.1 - Binomial Series for any rational n

[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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Covered syllabus points

P4.4 - Binomial expansion question 1