Worksheet: The Binomial Theorem

In this worksheet, we will practice on expanding any binomial expression of the form (a+b)^n.

Q1:

Use the binomial theorem to find the expansion of (1+๐‘ฅ)๏Šช.

  • A1+4๐‘ฅ+6๐‘ฅ+4๐‘ฅ+๐‘ฅ๏Šจ๏Šฉ๏Šช
  • B1+๐‘ฅ๏Šช
  • C1+4๐‘ฅ+6๐‘ฅ+4๐‘ฅ+4๐‘ฅ๏Šจ๏Šฉ๏Šช
  • D๐‘ฅ+4๐‘ฅ+4๐‘ฅ+๐‘ฅ๏Šจ๏Šฉ๏Šช
  • E1+3๐‘ฅ+6๐‘ฅ+10๐‘ฅ+15๐‘ฅ๏Šจ๏Šฉ๏Šช

Q2:

Expand (7+2๐‘ฅ)๏Šฉ.

  • Aโˆ’๐‘ฅ+21๐‘ฅโˆ’147๐‘ฅ+343๏Šฉ๏Šจ
  • B8๐‘ฅ+84๐‘ฅ+294๐‘ฅ+343๏Šฉ๏Šจ
  • Cโˆ’8๐‘ฅ+84๐‘ฅโˆ’294๐‘ฅ+343๏Šฉ๏Šจ
  • D๐‘ฅ+21๐‘ฅ+147๐‘ฅ+343๏Šฉ๏Šจ

Q3:

Use the binomial theorem to find the expansion of (๐‘Ž+2๐‘)๏Šช.

  • A๐‘Ž+8๐‘Ž๐‘+24๐‘Ž๐‘+32๐‘Ž๐‘+64๐‘๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • B๐‘Ž+4๐‘Ž๐‘+6๐‘Ž๐‘+4๐‘Ž๐‘+๐‘๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • C๐‘Ž+8๐‘Ž๐‘+24๐‘Ž๐‘+32๐‘Ž๐‘+16๐‘๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • D๐‘Ž+16๐‘๏Šช๏Šช
  • E๐‘Ž+4๐‘Ž๐‘+24๐‘Ž๐‘+32๐‘Ž๐‘+16๐‘๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช

Q4:

Use the binomial theorem to find the expansion of (๐‘Žโˆ’๐‘)๏Šซ.

  • A๐‘Žโˆ’5๐‘Ž๐‘+10๐‘Ž๐‘โˆ’10๐‘Ž๐‘+5๐‘Ž๐‘โˆ’๐‘๏Šซ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช๏Šซ
  • B๐‘Ž+5๐‘Ž๐‘+10๐‘Ž๐‘+10๐‘Ž๐‘+5๐‘Ž๐‘+๐‘๏Šซ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช๏Šซ
  • C๐‘Ž+5๐‘Ž๐‘โˆ’10๐‘Ž๐‘+10๐‘Ž๐‘โˆ’5๐‘Ž๐‘+๐‘๏Šซ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช๏Šซ
  • D๐‘Žโˆ’5๐‘Ž๐‘โˆ’10๐‘Ž๐‘โˆ’10๐‘Ž๐‘โˆ’5๐‘Ž๐‘โˆ’๐‘๏Šซ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช๏Šซ
  • E5๐‘Žโˆ’5๐‘Ž๐‘+10๐‘Ž๐‘โˆ’10๐‘Ž๐‘+5๐‘Ž๐‘โˆ’๐‘๏Šซ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช๏Šซ

Q5:

Find the third term in the expansion of ๏€ผ10๐‘ฅ+23๐‘ฅ๏ˆ๏Šจ๏Šช.

  • A4009๐‘ฅ๏Šช
  • B4009๐‘ฅ๏Šจ
  • C8003๐‘ฅ๏Šจ
  • D8003๐‘ฅ๏Šช

Q6:

Consider the expansion of ๏€ผ๐‘Ž๐‘ฅ+๐‘ฅ๏ˆ.๏Šช๏Šง๏Šฆ Given that the constant of this expansion is 720, find all the possible values of ๐‘Ž.

  • A4,โˆ’4
  • B2,โˆ’2
  • C16,โˆ’16
  • D8,โˆ’8

Q7:

Which of the following is equal to ๏Šง๏Šช๏Šง๏Šง๏Šช๏Šจ๏Šง๏Šช๏Šฉ๏Šง๏Šช๏Šง๏Šช๐ถ+2ร—๐ถ+3ร—๐ถ+โ‹ฏ+14ร—๐ถ?

  • A2๏Šง๏Šช
  • B14ร—2๏Šง๏Šฉ
  • C13ร—2๏Šง๏Šช
  • D2๏Šง๏Šฉ
  • E14ร—2๏Šง๏Šช

Q8:

Write the coefficients of the terms that result from the expansion of (๐‘ฅ+๐‘ฆ)๏Šช.

  • A1,5,10,5,1
  • B1,3,3,1
  • C1,2,1
  • D1,4,4,1
  • E1,4,6,4,1

Q9:

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

Q10:

Expand (๐‘ฅ+2๐‘ฆ)๏Šจ๏Šจ.

  • A๐‘ฅ+2๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šจ๏Šจ
  • B๐‘ฅ+4๐‘ฅ๐‘ฆ+4๐‘ฆ๏Šจ๏Šจ
  • C๐‘ฅ+2๐‘ฅ๐‘ฆ+๐‘ฆ๏Šจ๏Šจ
  • D๐‘ฅ+4๐‘ฅ๐‘ฆ+4๐‘ฆ๏Šช๏Šจ๏Šจ

Q11:

Evaluate ๏€ปโˆš3+1๏‡+๏€ปโˆš3โˆ’1๏‡๏Šฉ๏Šฉ using the binomial expansion theorem.

  • A36
  • B27โˆš3
  • C12
  • D27
  • E12โˆš3

Q12:

Expand ๏€ผ6๐‘ฅโˆ’13๐‘ฅ๏ˆ๏Šจ๏Šจ.

  • A36๐‘ฅ+4๐‘ฅ+19๐‘ฅ๏Šช๏Šจ
  • B36๐‘ฅโˆ’12๐‘ฅ+1๐‘ฅ๏Šช๏Šจ
  • C36๐‘ฅโˆ’4๐‘ฅ+19๐‘ฅ๏Šช๏Šจ
  • D๐‘ฅโˆ’2๐‘ฅ3+19๐‘ฅ๏Šช๏Šจ

Q13:

Expand ๏€ผ๐‘ฅ4โˆ’1๐‘ฅ๏ˆ๏Šซ.

  • A๐‘ฅ1,024โˆ’5๐‘ฅ256+5๐‘ฅ32โˆ’58๐‘ฅ+54๐‘ฅโˆ’1๐‘ฅ๏Šซ๏Šฉ๏Šฉ๏Šซ
  • B๐‘ฅโˆ’20๐‘ฅ+160๐‘ฅโˆ’640๐‘ฅ+1,280๐‘ฅโˆ’1,024๏Šง๏Šฆ๏Šฎ๏Šฌ๏Šช๏Šจ
  • C๐‘ฅโˆ’20๐‘ฅ+160๐‘ฅโˆ’640๐‘ฅ+1,280๐‘ฅโˆ’1,024๐‘ฅ๏Šง๏Šฆ๏Šญ๏Šช๏Šจ๏Šซ
  • D๐‘ฅโˆ’5๐‘ฅ+10๐‘ฅโˆ’10๐‘ฅ+5๐‘ฅโˆ’1๐‘ฅ๏Šซ๏Šฉ๏Šฉ๏Šซ

Q14:

Find the coefficient of the fourth term in the expansion of ๏€ผ๐‘ฅ+1๐‘ฅ๏ˆ๏Šช.

Q15:

Answer the following questions for the expansion of (2+๐‘˜๐‘ฅ)๏Šฌ.

Given that the coefficient of ๐‘ฅ๏Šจ is 60, and ๐‘˜ is positive, find ๐‘˜.

  • A๐‘˜=โˆš154
  • B๐‘˜=12
  • C๐‘˜=14
  • D๐‘˜=2
  • E๐‘˜=1

Hence, using your value of ๐‘˜, work out the coefficient of ๐‘ฅ๏Šซ in the expansion.

  • A12
  • B3256
  • C15
  • D384
  • E38

Q16:

Answer the following questions for the expansion of (1โˆ’3๐‘ฅ)๏Š.

Given that the coefficient of ๐‘ฅ๏Šจ is 189, find ๐‘›.

  • A๐‘›=10
  • B๐‘›=9
  • C๐‘›=7
  • D๐‘›=8
  • E๐‘›=6

Hence, work out the value of the coefficient of ๐‘ฅ๏Šซ.

Q17:

Consider the expansion of ๏€น๐‘ฅ+๐‘ฅ๏…๏Šฌ๏Šฑ๏Šฌ๏Šซ in descending powers of ๐‘ฅ. What are the possible values of it, if the third term in this expansion is equal to 640?

  • A12
  • B2,โˆ’2
  • C4,โˆ’4
  • D10

Q18:

Find the two middle terms in the expansion of (14๐‘ฅ+๐‘ฆ)๏Šฉ.

  • A588๐‘ฅ๐‘ฆ๏Šจ, 42๐‘ฅ๐‘ฆ๏Šจ
  • B2,744๐‘ฅ๐‘ฆ๏Šจ, 14๐‘ฅ๐‘ฆ๏Šจ
  • C196๐‘ฅ๐‘ฆ๏Šจ, 42๐‘ฅ๐‘ฆ๏Šจ
  • D196๐‘ฅ๐‘ฆ๏Šจ, 14๐‘ฅ๐‘ฆ๏Šจ

Q19:

Find ๐‘ฅ given that the ratio of the middle terms in the expansion of (1+๐‘ฅ)๏Šฉ is 1โˆถ2.

Q20:

Given that (1+๐‘๐‘ฅ)=1+6๐‘ฅ+๐‘Ž๐‘ฅ+๐‘Ž๐‘ฅ+โ‹ฏ+๐‘Ž๐‘ฅ๏Š๏Šง๏Šจ๏Šจ๏Šฉ๏Š๏Šฑ๏Šง๏Š and 2๐‘Ž=3๐‘Ž๏Šง๏Šจ, find the values of ๐‘› and ๐‘ where ๐‘โ‰ 0.

  • A๐‘›=4, ๐‘=3
  • B๐‘›=3, ๐‘=3
  • C๐‘›=3, ๐‘=2
  • D๐‘›=4, ๐‘=2

Q21:

Expand (5๐‘ฅ+4๐‘ฆ)๏Šช.

  • A๐‘ฅ+4๐‘ฅ๐‘ฆ+6๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • B625๐‘ฅ+2,000๐‘ฅ๐‘ฆ+2,400๐‘ฅ๐‘ฆ+1,280๐‘ฅ๐‘ฆ+256๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • C625๐‘ฅ+500๐‘ฅ๐‘ฆ+150๐‘ฅ๐‘ฆ+20๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • D๐‘ฅ+16๐‘ฅ๐‘ฆ+96๐‘ฅ๐‘ฆ+256๐‘ฅ๐‘ฆ+256๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช

Q22:

Expand ๏€ผ8๐‘ฅโˆ’74๐‘ฆ๏ˆ๏Šจ.

  • A64๐‘ฅโˆ’112๐‘ฅ๐‘ฆ+49๐‘ฆ๏Šจ๏Šจ
  • B64๐‘ฅ+28๐‘ฅ๐‘ฆ+49๐‘ฆ16๏Šจ๏Šจ
  • C64๐‘ฅโˆ’4๐‘ฅ๐‘ฆ+๐‘ฆ16๏Šจ๏Šจ
  • D64๐‘ฅโˆ’28๐‘ฅ๐‘ฆ+49๐‘ฆ16๏Šจ๏Šจ
  • E64๐‘ฅ+4๐‘ฅ๐‘ฆ+๐‘ฆ16๏Šจ๏Šจ

Q23:

Expand ๏€ป๐‘ฅโˆ’โˆš2๏‡๏Šฉ.

  • Aโˆ’๐‘ฅ+3โˆš2๐‘ฅโˆ’6๐‘ฅ+2โˆš2๏Šฉ๏Šจ
  • B๐‘ฅโˆ’3โˆš2๐‘ฅ+6๐‘ฅโˆ’2โˆš2๏Šฌ๏Šช๏Šจ
  • C๐‘ฅ+3โˆš2๐‘ฅ+6๐‘ฅ+2โˆš2๏Šฉ๏Šจ
  • D๐‘ฅโˆ’3โˆš2๐‘ฅ+6๐‘ฅโˆ’2โˆš2๏Šฉ๏Šจ

Q24:

Find the third term in the expansion of (4๐‘ฅ+3)๏Šฉ.

  • A27๐‘ฅ
  • B27๐‘ฅ๏Šจ
  • C108๐‘ฅ
  • D108๐‘ฅ๏Šจ

Q25:

Find the value of ๐‘ฅ that satisfies 1+9๐‘ฅ+9ร—82ร—1๐‘ฅ+9ร—8ร—73ร—2ร—1๐‘ฅ+โ‹ฏ+๐‘ฅ=512.๏Šจ๏Šฉ๏Šฏ

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