Lesson Worksheet: The Binomial Theorem Mathematics

In this worksheet, we will practice using Pascal's triangle to find the coefficients of expansion of (a+b)n and using the binomial theorem to find the expansion of (a+b)n.


What is the middle term in the following expansion: 1+๐ถ๐‘ฅ+๐ถ๐‘ฅ+โ‹ฏ+๐ถ๐‘ฅ.๏Šง๏Šฌ๏Šง๏Šง๏Šฌ๏Šจ๏Šจ๏Šง๏Šฌ๏Šง๏Šฌ๏Šง๏Šฌ

  • A11440๐‘ฅ๏Šฏ
  • B12870๐‘ฅ๏Šง๏Šฆ
  • C11440๐‘ฅ๏Šญ
  • D12870๐‘ฅ๏Šฎ


Find the middle term in the expansion of (15๐‘Ž+3๐‘)๏Šจ.

  • Aโˆ’45๐‘Ž๐‘
  • Bโˆ’90๐‘Ž๐‘
  • C45๐‘Ž๐‘
  • D90๐‘Ž๐‘


In the expansion of (2+๐‘ฅ)๏Šฌ according to the descending powers of ๐‘ฅ, if the value of the fourth term is 4โ€Žโ€‰โ€Ž320, find the value of ๐‘ฅ.

  • A3
  • B3โˆš2
  • C6โˆš2
  • D27
  • E18


In the expansion of ๏€ผ2๐‘ฅโˆ’๐‘ฅ๏ˆ๏Šจ๏Šฎ according to the descending powers of ๐‘ฅ, find the fourth term.

  • Aโˆ’448๐‘ฅ๏Šจ
  • Bโˆ’448๐‘ฅ
  • C1120๐‘ฅ๏Šช
  • D112๐‘ฅ๏Šจ
  • E112๐‘ฅ๏Šช


In the expansion of ๏€ผ๐‘ฅโˆ’12๏ˆ๏Šจ๏Šฎ according to the descending powers of ๐‘ฅ, find the second term from the end.

  • Aโˆ’4๐‘ฅ๏Šง๏Šช
  • B7๐‘ฅ๏Šง๏Šจ
  • Cโˆ’๐‘ฅ16๏Šจ
  • D7๐‘ฅ16๏Šช
  • E๐‘ฅ16๏Šช


Use Pascalโ€™s triangle to expand the expression ๏€ผ๐‘ฅ+1๐‘ฅ๏ˆ๏Šช.

  • A๐‘ฅ+4๐‘ฅ+6+4๐‘ฅ+1๐‘ฅ๏Šช๏Šจ๏Šจ๏Šช
  • B๐‘ฅ+4๐‘ฅ+6+4๐‘ฅ+1๐‘ฅ๏Šช๏Šฉ๏Šช
  • C๐‘ฅ+4๐‘ฅ+6+4๐‘ฅ+1๐‘ฅ๏Šช๏Šฉ๏Šจ๏Šช
  • D๐‘ฅ+4๐‘ฅ+6+1๐‘ฅ+1๐‘ฅ๏Šช๏Šจ๏Šจ๏Šช
  • E๐‘ฅ+6๐‘ฅ+6+4๐‘ฅ+1๐‘ฅ๏Šช๏Šจ๏Šจ๏Šช


Use Pascalโ€™s triangle to expand the expression (3+๐‘ฅ)๏Šช.

  • A๐‘ฅ+12๐‘ฅ+54๐‘ฅ+108๐‘ฅ๏Šช๏Šฉ๏Šจ
  • B๐‘ฅ+12๐‘ฅ+54๐‘ฅ+90๐‘ฅ+81๏Šช๏Šฉ๏Šจ
  • C๐‘ฅ+12๐‘ฅ+54๐‘ฅ+108๐‘ฅ+81๏Šช๏Šฉ๏Šจ
  • D๐‘ฅ+9๐‘ฅ+81๐‘ฅ+81๐‘ฅ+81๏Šช๏Šฉ๏Šจ
  • E๐‘ฅ+4๐‘ฅ+18๐‘ฅ+36๐‘ฅ+27๏Šช๏Šฉ๏Šจ


Use Pascalโ€™s triangle to expand the expression (๐‘ฅ+๐‘ฆ)๏Šช.

  • A๐‘ฅ+3๐‘ฅ๐‘ฆ+6๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • B๐‘ฅ+3๐‘ฅ๐‘ฆ+9๐‘ฅ๐‘ฆ+3๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • C๐‘ฅ+4๐‘ฅ๐‘ฆ+9๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • D๐‘ฅ+4๐‘ฅ๐‘ฆ+6๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šจ๏Šจ๏Šจ๏Šฉ๏Šช
  • E๐‘ฅ+4๐‘ฅ๐‘ฆ+6๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช


Use the binomial expansion to find the first three terms of (2๐‘Žโˆ’2)๏Šญ in ascending powers of ๐‘Ž.

  • A128๐‘Žโˆ’896๐‘Ž+2688๐‘Ž๏Šญ๏Šฌ๏Šซ
  • Bโˆ’128+896๐‘Žโˆ’2688๐‘Ž๏Šจ
  • Cโˆ’128+896๐‘Ž+2688๐‘Ž๏Šจ
  • D128๐‘Ž+896๐‘Ž+2688๐‘Ž๏Šญ๏Šฌ๏Šซ
  • Eโˆ’128+128๐‘Žโˆ’2688๐‘Ž๏Šจ


Use the binomial theorem to expand (2๐‘ฅโˆ’3๐‘ฆ)๏Šฉ.

  • A8๐‘ฅโˆ’12๐‘ฅ๐‘ฆ+18๐‘ฅ๐‘ฆโˆ’27๐‘ฆ๏Šฉ๏Šจ๏Šจ๏Šฉ
  • B8๐‘ฅ+36๐‘ฅ๐‘ฆ+54๐‘ฅ๐‘ฆ+27๐‘ฆ๏Šฉ๏Šจ๏Šจ๏Šฉ
  • C8๐‘ฅ+36๐‘ฅ๐‘ฆโˆ’54๐‘ฅ๐‘ฆ+27๐‘ฆ๏Šฉ๏Šจ๏Šจ๏Šฉ
  • D8๐‘ฅโˆ’36๐‘ฅ๐‘ฆ+54๐‘ฅ๐‘ฆโˆ’27๐‘ฆ๏Šฉ๏Šจ๏Šจ๏Šฉ
  • E8๐‘ฅโˆ’36๐‘ฅ๐‘ฆโˆ’54๐‘ฅ๐‘ฆโˆ’27๐‘ฆ๏Šฉ๏Šจ๏Šจ๏Šฉ

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