The Binomial Expansion · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Expanding brackets without the pain

Mathematics · Algebra · ages 16-17
Name ______________________   Date ____________

  1. In the expansion of (1 + 2x) to the power 5, what is the coefficient of x squared?

    Answer: ______________

  2. How many terms appear in a full expansion of a power of 5?

    • 5
    • 10
    • 6

  3. In each term the powers of the second bracket climb from 0 to n.

    Circle one:   True   False

  4. Which row of the triangle serves 1 plus 2x to the 5?

    • 1, 5, 10, 10, 5, 1
    • 1, 4, 6, 4, 1
    • 1, 5, 10, 5, 1

  5. Use the first three terms of the binomial expansion to approximate 1.02 to the power 8.

    Answer: ______________

  6. In the expansion of (1 + 2x) to the power 5, what is the coefficient of x cubed?

    • 80
    • 40
    • 20
    • 160

  7. Using the first three terms of 1.02 to the 8 gives an exact answer.

    Circle one:   True   False

  8. What is the coefficient of x cubed in 1 plus 2x to the 5?

    Answer: ______________

  9. Why does the three-term estimate of 1.02 to the 8 fall short?

    • The first terms are always negative
    • Dropped terms are positive, so the true value runs higher
    • 0.02 was rounded up

  10. In the expansion of (2 minus x) to the power 6, what is the coefficient of x squared?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Expanding brackets without the pain W1-mt_z16HZkNZ8c-s1

  1. 40 · The x squared term is 5 choose 2 times (2x) squared = 10 times 4x squared, so the coefficient is 40.
  2. 6 · A power of n always gives n plus 1 terms.
  3. True · The second term climbs while the first one falls.
  4. 1, 5, 10, 10, 5, 1 · Row 5 lists the coefficients for the fifth power.
  5. 1.1712 · 1.02 to the 8 is (1 + 0.02) to the 8, and the first three terms give 1 + 0.16 + 0.0112 = 1.1712.
  6. 80 · The x cubed term is 5 choose 3 times (2x) cubed = 10 times 8x cubed, so the coefficient is 80.
  7. False · Dropped terms are small but real, so the result is an estimate.
  8. 80 · Take 10 times 2 cubed, which is 10 times 8.
  9. Dropped terms are positive, so the true value runs higher · Every omitted term adds a little more, all positive here.
  10. 240 · The x squared term is 6 choose 2 times 2 to the 4 times (negative x) squared = 15 times 16 x squared, so the coefficient is 240.
Worksheet · LightMySky