4.2PureStretch

Expanding (a + bx)ⁿ

To expand $(a + b x)^{n}$ 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 Video by Zeeshan Zamurred Binomial Expansion

Edexcel A Level Maths: 4.2 Binomial ExpansionWatch the full walkthrough 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 $(a + b x)^{n} = a^{n} (1 + \frac{b}{a} x)^{n}$ . Now the bracket is in $(1 + X)^{n}$ form with $X = \frac{b}{a} x$ , and the constant $a^{n}$ multiplies every term.

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

Factor a out first — never expand directly.

Adjusted validity

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

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

Worked example

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

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

= 2 [1 + \frac{1}{2} \cdot \frac{x}{4} + \frac{( 1/2 ) ( - 1/2 )}{2} (\frac{x}{4})^{2} + \dots]

= 2 + \frac{x}{4} - \frac{x ^{2}}{64} + \dots

Answer

2 + \frac{x}{4} - \frac{x ^{2}}{64} + \dots

, valid for

∣ x ∣ < 4

Formula recap

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

Factor a out first.

∣ x ∣ < \frac{a}{b}

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|.

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