How do you expand (a+bx)^n?

Write it as a^n\left(1+\frac{b}{a}x\right)^n and expand with \frac{b}{a}x as the variable.

Write \sqrt{4+x} in the form needed for expansion.

2\left(1+\frac{x}{4}\right)^{\frac12}

Why must x be inside the validity range when approximating?

Outside it the series does not converge, so the estimate is meaningless.

Common error when taking a factor out of (a+bx)^n?

Forgetting to raise a to the power n.

Percentage error in an estimate?

\frac{estimate-exact}{exact}\times100\%

Binomial expansion for rational nAQA A-Level Maths: Flashcards

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Binomial expansion of $(1+x)^n$ for rational $n$?

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Binomial expansion of $(1+x)^n$ for rational $n$ ?: $1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\dots$
For which $x$ is the expansion of $(1+x)^n$ valid when $n$ is not a positive integer?: $|x|<1$
Does the expansion stop if $n$ is negative or fractional?: No, it is an infinite series.
How do you expand $(a+bx)^n$ ?: Write it as $a^n\left(1+\frac{b}{a}x\right)^n$ and expand with $\frac{b}{a}x$ as the variable.
Validity of the expansion of $(a+bx)^n$ ?: $\left|\frac{bx}{a}\right|<1$
Expand $(1+3x)^{-2}$ up to $x^2$ .: $1-6x+27x^2$
Write $\sqrt{4+x}$ in the form needed for expansion.: $2\left(1+\frac{x}{4}\right)^{\frac12}$
Range of validity for $(2+5x)^{-1}$ ?: $|x|<\frac25$
Why must $x$ be inside the validity range when approximating?: Outside it the series does not converge, so the estimate is meaningless.
Expand $(1-2x)^{-\frac12}$ up to $x^2$ .: $1+x+\frac32x^2$
Common error when taking a factor out of $(a+bx)^n$ ?: Forgetting to raise $a$ to the power $n$ .
Percentage error in an estimate?: $\frac{\text{estimate}-\text{exact}}{\text{exact}}\times100\%$

Exam questions on Binomial expansion for rational n

The function $f(x)=(1+3x)^{-2}$ is expanded in ascending powers of $x$ .
Use the first three terms of the expansion, with a suitable value of $x$ , to estimate $1.03^{-2}$ .2 marks
Let $g(x)=\sqrt{4+x}$ .
Find the first three terms of the expansion of $g(x)$ in ascending powers of $x$ .2 marks
Let $f(x)=\frac{1}{2+5x}$ .
Find the expansion of $f(x)$ in ascending powers of $x$ , up to and including the term in $x^2$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).