All subjects · Mathematics · WMA12 (P2)

Binomial expansion: unknowns and approximations

Exam questions often give you a coefficient and ask for an unknown, or use the first few terms to estimate a number.

Unknown constants

Write the general term with the unknown, set it equal to the given coefficient, and solve.

(1 + kx)6: the x2 term is 6C2 (kx)2 = 15k2 x2.

Products

To find a coefficient in (1 + x)(1 + 2x)5, combine the terms that multiply to give that power. For x2: 1 x (x2 term) + x x (x term).

Approximations

Choose x so the bracket matches the number: 1.0258 = (1 + x)8 with x = 0.025. When x is small, x2, x3, ... are very small, so the first few terms give a good estimate.

Worked example

The coefficient of x2 in (1 + kx)6 is 135. Find the possible values of k.

  1. The x2 term is 6C2 (kx)2 = 15k2 x2
  2. 15k2 = 135
  3. k2 = 9

Answer: k = 3 or k = -3

Key idea: Set the general term equal to the given coefficient. For small x, the first few terms approximate (1 + x)n well.

Check yourself

  1. The coefficient of x in (1 + kx)5 is 15. Find k.
    Show answer

    3. The x term is 5kx, so 5k = 15 and k = 3.

  2. The coefficient of x2 in (1 + ax)8 is 112, and a > 0. Find a.
    Show answer

    2. 8C2 a2 = 28a2 = 112, so a2 = 4 and a = 2.

  3. Use the first three terms of (1 + x)8 to estimate 1.0258.
    Show answer

    1.2175. 1 + 8(0.025) + 28(0.025)2 = 1 + 0.2 + 0.0175 = 1.2175.

  4. Find the coefficient of x2 in (1 + x)(1 + 2x)5.
    Show answer

    50. (1 + 2x)5 = 1 + 10x + 40x2 + ... The x2 terms are 1 x 40x2 and x x 10x, giving 50.

  5. In the expansion of (1 + px)6, p > 0, the coefficient of x2 is twice the coefficient of x. Find p.
    Show answer

    0.8. 15p2 = 2 x 6p, so 15p = 12 and p = 0.8.

  6. Why does a binomial approximation work best when x is small?
    • The coefficients get smaller
    • The expansion stops after three terms
    • Small x makes n a positive integer
    • The ignored terms in x3, x4, ... are tiny
    Show answer

    The ignored terms in x3, x4, ... are tiny. Powers of a small number shrink quickly, so leaving them out changes the answer very little.

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