Expanding (a + bx)ⁿ and rational functions

Taking out aⁿ

(a + bx)ⁿ = aⁿ(1 + (b/a)x)ⁿ. The whole of a is raised to the power n:

(4 + x)^1/2 = 4^1/2(1 + x/4)^1/2 = 2(1 + x/4)^1/2.

The expansion is valid for |bx/a| < 1, that is |x| < |a/b|.

Rational functions

To expand something like 3/((1 + x)(1 - 2x)):

1. Split into partial fractions: 1/(1 + x) + 2/(1 - 2x).

2. Expand each as a binomial series.

3. Add like terms.

The result is valid where both expansions are valid: the narrower range.

Common slips

• Writing 4(1 + x/4)^1/2 instead of 2(1 + x/4)^1/2.
• Forgetting to square x/4 in the x² term.
• Sign errors with (1 - 2x)^-1 = 1 + 2x + 4x² + ...

Worked example

Expand sqrt(4 + x) up to the x² term.

1. sqrt(4 + x) = 2(1 + x/4)^1/2
2. (1 + x/4)^1/2 = 1 + (1/2)(x/4) + (-1/8)(x/4)² = 1 + x/8 - x²/128
3. Multiply by 2.

Answer: 2 + x/4 - x²/64, valid for |x| < 4

Check yourself

1. Which is the correct first step to expand (9 + x)^1/2?
   • 9(1 + x/9)^1/2
   • 3(1 + x)^1/2
   • (1/3)(1 + x/9)^1/2
   • 3(1 + x/9)^1/2
   Show answer

   3(1 + x/9)^1/2. 9^1/2 = 3, and 9 + x = 9(1 + x/9).

2. Find the coefficient of x in the expansion of (2 + x)^-1. Give your answer as a fraction in its simplest form.
   Show answer

   -1/4. (2 + x)^-1 = (1/2)(1 + x/2)^-1 = (1/2)(1 - x/2 + ...). The x coefficient is -1/4.

3. Expand (4 - x)^-1/2 in ascending powers of x up to x². What is the x² coefficient? Give your answer as a fraction in its simplest form.
   Show answer

   3/256. (4 - x)^-1/2 = (1/2)(1 - x/4)^-1/2. The x² term inside is ((-1/2)(-3/2)/2)(-x/4)² = (3/8)(x²/16) = 3x²/128. Multiply by 1/2: 3/256.

4. For which values of x is the expansion of (3 + 2x)^-2 valid?
   • |x| < 3/2
   • |x| < 2/3
   • |x| < 1
   • |x| < 3
   Show answer

   |x| < 3/2. (3 + 2x)^-2 = (1/9)(1 + 2x/3)^-2, valid for |2x/3| < 1, so |x| < 3/2.

5. 3/((1 + x)(1 - 2x)) ≡ 1/(1 + x) + 2/(1 - 2x). Find the coefficient of x² in its series expansion.
   Show answer

   9. 1/(1 + x) = 1 - x + x² - ... and 2/(1 - 2x) = 2(1 + 2x + 4x² + ...) = 2 + 4x + 8x² + ... Adding: 3 + 3x + 9x² + ...

6. For which values of x is the expansion of 1/(1 + x) + 2/(1 - 2x) valid?
   • |x| < 1/2
   • |x| < 1
   • |x| < 2
   • |x| < 3/2
   Show answer

   |x| < 1/2. 1/(1 + x) needs |x| < 1 and 2/(1 - 2x) needs |x| < 1/2. Both must hold, so |x| < 1/2.

7. sqrt(4 + x) ≈ 2 + x/4 - x²/64. Use this with x = 0.1 to estimate sqrt(4.1), to 5 decimal places.
   Show answer

   2.02484. 2 + 0.025 - 0.00015625 = 2.02484375, which is 2.02484 to 5 decimal places.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 4 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.