The Binomial Expansion · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Expanding brackets without the pain

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

  1. What is the constant term in the expansion of (2 + x)^4?

    Answer: ______________

  2. Before substituting values for a and b, how many symbolic terms are in the binomial expansion of (a + b)^5?

    • 5
    • 10
    • 6

  3. For integer n at least 0, when (a + b)^n is expanded with powers of a decreasing, powers of b increase from 0 to n.

    Circle one:   True   False

  4. With the top of Pascal's triangle numbered row 0, which row gives the binomial coefficients for (1 + 2x)^5?

    • 1, 5, 10, 10, 5, 1
    • 1, 5, 10, 15, 5, 1
    • 1, 4, 10, 10, 4, 1

  5. Use the first three terms, ordered by increasing powers of h, in (1 + h)^6 to approximate 1.01^6. Give the three-term sum without further rounding.

    Answer: ______________

  6. What is the coefficient of x^3 in (2 - 3x)^4?

    • -216
    • -108
    • 108
    • 216

  7. The first three terms, ordered by increasing powers of h, in (1 + h)^8 are 8h, 28h^2 and 56h^3.

    Circle one:   True   False

  8. What is the coefficient of x^3 in (1 + 2x)^5?

    Answer: ______________

  9. Use the terms with k = 0, 1 and 2 to estimate (1 + 0.02)^8. Which statement about the omitted terms and the estimate is correct?

    • The omitted terms are all negative, so the estimate exceeds the exact value.
    • The omitted terms are positive, so adding them increases the estimate.
    • The omitted terms cancel in pairs, so the estimate equals the exact value.

  10. Write the terms of (x^2 + 3)^6 in decreasing powers of x, counting the first term as term 1. What is the term number of the term containing x^8?

    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. 16 · The constant term uses k = 0. It is (4 choose 0) × 2^4 × x^0 = 16.
  2. 6 · The index k takes the six values 0, 1, 2, 3, 4 and 5, giving six symbolic terms.
  3. True · Each term contains a^(n-k)b^k. Increasing k decreases a's exponent and increases b's exponent.
  4. 1, 5, 10, 10, 5, 1 · Exponent 5 uses row 5. Its entries are 1, 5, 10, 10, 5 and 1.
  5. 1.0615 · With h = 0.01, the first three terms give 1 + 6(0.01) + 15(0.01)^2 = 1.0615.
  6. -216 · Use k = 3: (4 choose 3) × 2^(4-3) × (-3)^3 = 4 × 2 × (-27) = -216.
  7. False · The first three terms correspond to k = 0, 1 and 2. They are 1, 8h and 28h^2.
  8. 80 · Use k = 3: (5 choose 3) × 1^2 × 2^3 = 10 × 8 = 80.
  9. The omitted terms are positive, so adding them increases the estimate. · Every omitted term is a positive binomial coefficient times a positive power of 0.02. Their sum raises the exact value above the estimate.
  10. 3 · Term k contains x^(2(6-k)). Solving 2(6-k) = 8 gives k = 2, so its term number is k + 1 = 3.
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