4.2PureStretch

Expanding (a + bx)ⁿ

To expand $(a+bx)^n$ for rational $n$ you must first factor out $a$ so the bracket starts with 1. Then apply the standard $(1+X)^n$ expansion — and adjust the range of validity accordingly.

30 min Binomial Expansion

Edexcel A Level Maths: 4.2 Binomial ExpansionCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Factor (a + bx)ⁿ into aⁿ(1 + (b/a)x)ⁿ
Apply the general binomial expansion
Determine the adjusted range of validity
Expand expressions with rational powers

Factor out a

Write $(a+bx)^n = a^n\left(1 + \tfrac{b}{a}x\right)^n$ . Now the bracket is in $(1+X)^n$ form with $X = \tfrac{b}{a}x$ , and the constant $a^n$ multiplies every term.

(a+bx)^n = a^n\left(1 + \tfrac{b}{a}x\right)^n

Factor a out first — never expand directly.

Adjusted validity

The expansion of $\left(1+\tfrac{b}{a}x\right)^n$ is valid when $\left|\tfrac{b}{a}x\right| < 1$ , i.e. $|x| < \left|\tfrac{a}{b}\right|$ .

Tip — Validity widens or narrows depending on b/a — always recompute it.

Worked example

Expand $(4+x)^{1/2}$ up to the term in $x^2$ .

Example 1Find the first three terms of

\sqrt{4+x}

= 4^{1/2}(1 + \tfrac{x}{4})^{1/2} = 2(1+\tfrac{x}{4})^{1/2}

= 2\left[1 + \tfrac12\cdot\tfrac{x}{4} + \tfrac{(1/2)(-1/2)}{2}\left(\tfrac{x}{4}\right)^2 + \cdots\right]

= 2 + \tfrac{x}{4} - \tfrac{x^2}{64} + \cdots

Answer

2 + \tfrac{x}{4} - \tfrac{x^2}{64} + \cdots

, valid for

|x| < 4

Formula recap

(a+bx)^n = a^n(1 + \tfrac{b}{a}x)^n

Factor a out first.

|x| < \left|\tfrac{a}{b}\right|

Adjusted validity.

Common mistakes to avoid

Expanding (a + bx)ⁿ without factoring out a.

The (1+X)ⁿ formula needs the bracket to start with 1 — factor a first.

Keeping validity as |x| < 1.

Validity is |x| < |a/b| after factoring.

Key takeaways

Factor: (a+bx)ⁿ = aⁿ(1 + (b/a)x)ⁿ.
Apply the standard expansion to the bracket, multiply by aⁿ.
Validity becomes |x| < |a/b|.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

To expand $(a+bx)^n$ for rational $n$ , first:
- factor out $a$
- factor out $b$
- use ⁿCᵣ
- square it
Show answer and explanation
Answer: factor out $a$
Bracket must start with 1.
$(4+x)^{1/2}$ equals:
- $2(1+\tfrac{x}{4})^{1/2}$
- $4(1+\tfrac{x}{4})^{1/2}$
- $2(1+x)^{1/2}$
- $(1+\tfrac{x}{4})^{1/2}$
Show answer and explanation
Answer: $2(1+\tfrac{x}{4})^{1/2}$
4^{1/2}=2.
The validity of $(4+x)^{1/2}$ is:
- $|x| < 4$
- $|x| < 1$
- $|x| < \tfrac14$
- all $x$
Show answer and explanation
Answer: $|x| < 4$
|x| < |a/b| = 4.
The constant term of $\sqrt{4+x}$ is:
- $2$
- $4$
- $1$
- $\tfrac12$
Show answer and explanation
Answer: $2$
aⁿ·1 = 2.
For $(a+bx)^n$ , the range of validity is:
- $|x| < |a/b|$
- $|x| < |b/a|$
- $|x| < 1$
- $|x| < ab$
Show answer and explanation
Answer: $|x| < |a/b|$
|(b/a)x| < 1.

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