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Binomial Theorem Practice Problems

Use Pascal’s triangle or nCk for the coefficients, and count the powers down on the first term and up on the second.

  1. 1.

    Evaluate .

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    .

  2. 2.

    Evaluate .

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    .

  3. 3.

    Expand .

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    Row 3 of Pascal’s triangle is 1, 3, 3, 1.

  4. 4.

    Write row 6 of Pascal’s triangle (the coefficients of ), separated by commas.

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    Each entry is the sum of the two above it in row 5 (1, 5, 10, 10, 5, 1).

  5. 5.

    Expand .

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    Coefficients 1, 4, 6, 4, 1 times powers of 2: .

  6. 6.

    Expand .

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    .

  7. 7.

    Find the coefficient of in the expansion of .

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    The term is .

  8. 8.

    What is the coefficient of the 4th term of ?

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    The 4th term uses : , giving .

  9. 9.

    Find the coefficient of in .

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    .

  10. 10.

    Find the constant term in the expansion of .

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    The general term is , whose power of is . That is 0 when : .

  11. 11.

    Find the coefficient of in .

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    .

  12. 12.

    Use the first three terms of the binomial expansion to estimate .

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    . The exact value is 1.05101…
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Binomial Theorem Practice Problems

These problems follow the binomial theorem lesson. Type expansions with ^ for powers, for example x^3 + 3x^2 + 3x + 1; any equivalent form is accepted.

The theorem

Common mistakes to watch for

  • Forgetting the coefficients. has binomial coefficients in front of each term, not just powers.
  • Off-by-one term numbers. The th term uses : the first term is .
  • Dropping signs. For , write it as so the signs alternate correctly.
  • Not raising the coefficient. In , the 2 is cubed too: .

Frequently Asked Questions

How do you find a specific term without expanding everything?+

The term with b^k is C(n, k) · a^(n − k) · b^k. Choose k to give the power you need.

What is the sum of the coefficients of (x + 1)^n?+

Set x = 1: the sum is 2^n. Row n of Pascal’s triangle always adds to 2^n.

How do I type my answers?+

Type numbers without units or commas, and fractions like 7/9. For formulas, use n and ^ for powers, such as 3n - 1 or 5*2^(n - 1).

Calculators for this topic

Binomial Expansion Calculator

More practice problems

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