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 + 𝑥 ) 4 .

  • A 𝑥 + 4 𝑥 + 4 𝑥 + 𝑥 2 3 4
  • B 1 + 𝑥 4
  • C 1 + 3 𝑥 + 6 𝑥 + 1 0 𝑥 + 1 5 𝑥 2 3 4
  • D 1 + 4 𝑥 + 6 𝑥 + 4 𝑥 + 𝑥 2 3 4
  • E 1 + 4 𝑥 + 6 𝑥 + 4 𝑥 + 4 𝑥 2 3 4

Q2:

Expand ( 7 + 2 𝑥 ) .

  • A 8 𝑥 + 8 4 𝑥 2 9 4 𝑥 + 3 4 3
  • B 𝑥 + 2 1 𝑥 + 1 4 7 𝑥 + 3 4 3
  • C 𝑥 + 2 1 𝑥 1 4 7 𝑥 + 3 4 3
  • D 8 𝑥 + 8 4 𝑥 + 2 9 4 𝑥 + 3 4 3

Q3:

Use the binomial theorem to find the expansion of ( 𝑎 + 2 𝑏 ) 4 .

  • A 𝑎 + 1 6 𝑏 4 4
  • B 𝑎 + 4 𝑎 𝑏 + 2 4 𝑎 𝑏 + 3 2 𝑎 𝑏 + 1 6 𝑏 4 3 2 2 3 4
  • C 𝑎 + 8 𝑎 𝑏 + 2 4 𝑎 𝑏 + 3 2 𝑎 𝑏 + 6 4 𝑏 4 3 2 2 3 4
  • D 𝑎 + 8 𝑎 𝑏 + 2 4 𝑎 𝑏 + 3 2 𝑎 𝑏 + 1 6 𝑏 4 3 2 2 3 4
  • E 𝑎 + 4 𝑎 𝑏 + 6 𝑎 𝑏 + 4 𝑎 𝑏 + 𝑏 4 3 2 2 3 4

Q4:

Use the binomial theorem to find the expansion of ( 𝑎 𝑏 ) 5 .

  • A 5 𝑎 5 𝑎 𝑏 + 1 0 𝑎 𝑏 1 0 𝑎 𝑏 + 5 𝑎 𝑏 𝑏 5 4 3 2 2 3 4 5
  • B 𝑎 + 5 𝑎 𝑏 + 1 0 𝑎 𝑏 + 1 0 𝑎 𝑏 + 5 𝑎 𝑏 + 𝑏 5 4 3 2 2 3 4 5
  • C 𝑎 5 𝑎 𝑏 1 0 𝑎 𝑏 1 0 𝑎 𝑏 5 𝑎 𝑏 𝑏 5 4 3 2 2 3 4 5
  • D 𝑎 5 𝑎 𝑏 + 1 0 𝑎 𝑏 1 0 𝑎 𝑏 + 5 𝑎 𝑏 𝑏 5 4 3 2 2 3 4 5
  • E 𝑎 + 5 𝑎 𝑏 1 0 𝑎 𝑏 + 1 0 𝑎 𝑏 5 𝑎 𝑏 + 𝑏 5 4 3 2 2 3 4 5

Q5:

Find the third term in the expansion of 1 0 𝑥 + 2 3 𝑥 .

  • A 4 0 0 9 𝑥
  • B 8 0 0 3 𝑥
  • C 4 0 0 9 𝑥
  • D 8 0 0 3 𝑥

Q6:

Consider the expansion of 𝑎 𝑥 + 𝑥 . Given that the constant of this expansion is 720, find all the possible values of 𝑎 .

  • A 2 , 2
  • B 1 6 , 1 6
  • C 8 , 8
  • D 4 , 4

Q7:

Which of the following is equal to 𝐶 + 2 × 𝐶 + 3 × 𝐶 + + 1 4 × 𝐶 ?

  • A 2
  • B 1 4 × 2
  • C 2
  • D 1 4 × 2
  • E 1 3 × 2

Q8:

Write the coefficients of the terms that result from the expansion of ( 𝑥 + 𝑦 ) 4 .

  • A 1 , 4 , 4 , 1
  • B 1 , 5 , 1 0 , 5 , 1
  • C 1 , 3 , 3 , 1
  • D 1 , 4 , 6 , 4 , 1
  • E 1 , 2 , 1

Q9:

Use the binomial theorem to expand ( 2 𝑥 3 𝑦 ) 3 .

  • A 8 𝑥 + 3 6 𝑥 𝑦 + 5 4 𝑥 𝑦 + 2 7 𝑦 3 2 2 3
  • B 8 𝑥 + 3 6 𝑥 𝑦 5 4 𝑥 𝑦 + 2 7 𝑦 3 2 2 3
  • C 8 𝑥 3 6 𝑥 𝑦 5 4 𝑥 𝑦 2 7 𝑦 3 2 2 3
  • D 8 𝑥 3 6 𝑥 𝑦 + 5 4 𝑥 𝑦 2 7 𝑦 3 2 2 3
  • E 8 𝑥 1 2 𝑥 𝑦 + 1 8 𝑥 𝑦 2 7 𝑦 3 2 2 3

Q10:

Expand ( 𝑥 + 2 𝑦 ) .

  • A 𝑥 + 2 𝑥 𝑦 + 𝑦
  • B 𝑥 + 4 𝑥 𝑦 + 4 𝑦
  • C 𝑥 + 2 𝑥 𝑦 + 𝑦
  • D 𝑥 + 4 𝑥 𝑦 + 4 𝑦

Q11:

Evaluate 3 + 1 + 3 1 using the binomial expansion theorem.

  • A36
  • B12
  • C 2 7 3
  • D 1 2 3
  • E27

Q12:

Expand 6 𝑥 1 3 𝑥 .

  • A 𝑥 2 𝑥 3 + 1 9 𝑥
  • B 3 6 𝑥 1 2 𝑥 + 1 𝑥
  • C 3 6 𝑥 + 4 𝑥 + 1 9 𝑥
  • D 3 6 𝑥 4 𝑥 + 1 9 𝑥

Q13:

Expand 𝑥 4 1 𝑥 .

  • A 𝑥 2 0 𝑥 + 1 6 0 𝑥 6 4 0 𝑥 + 1 , 2 8 0 𝑥 1 , 0 2 4
  • B 𝑥 5 𝑥 + 1 0 𝑥 1 0 𝑥 + 5 𝑥 1 𝑥
  • C 𝑥 2 0 𝑥 + 1 6 0 𝑥 6 4 0 𝑥 + 1 , 2 8 0 𝑥 1 , 0 2 4 𝑥
  • D 𝑥 1 , 0 2 4 5 𝑥 2 5 6 + 5 𝑥 3 2 5 8 𝑥 + 5 4 𝑥 1 𝑥

Q14:

Find the coefficient of 𝑥 in the expansion of 1 + 𝑥 𝑥 .

Q15:

Find the coefficient of the fourth term in the expansion of 𝑥 + 1 𝑥 .

  • A14
  • B8
  • C6
  • D4

Q16:

Answer the following questions for the expansion of ( 2 + 4 𝑥 ) .

Given that the coefficient of 𝑥 is 3 8 4 0 , find 𝑛 .

  • A 𝑛 = 8
  • B 𝑛 = 7
  • C 𝑛 = 5
  • D 𝑛 = 6
  • E 𝑛 = 9

Hence, work out the value of the coefficient of 𝑥 .

Q17:

Answer the following questions for the expansion of ( 2 + 𝑘 𝑥 ) 6 .

Given that the coefficient of 𝑥 2 is 60, and 𝑘 is positive, find 𝑘 .

  • A 𝑘 = 1
  • B 𝑘 = 2
  • C 𝑘 = 1 4
  • D 𝑘 = 1 2
  • E 𝑘 = 1 5 4

Hence, using your value of 𝑘 , work out the coefficient of 𝑥 5 in the expansion.

  • A 3 8
  • B 3 2 5 6
  • C12
  • D384
  • E 1 5

Q18:

Answer the following questions for the expansion of ( 1 3 𝑥 ) .

Given that the coefficient of 𝑥 is 189, find 𝑛 .

  • A 𝑛 = 6
  • B 𝑛 = 1 0
  • C 𝑛 = 9
  • D 𝑛 = 7
  • E 𝑛 = 8

Hence, work out the value of the coefficient of 𝑥 .

Q19:

Consider the expansion of 𝑥 + 𝑥 in descending powers of 𝑥 . What are the possible values of it, if the third term in this expansion is equal to 640?

  • A10
  • B12
  • C 4 , 4
  • D 2 , 2

Q20:

Find the two middle terms in the expansion of ( 1 4 𝑥 + 𝑦 ) .

  • A 2 7 4 4 𝑥 𝑦 , 1 4 𝑥 𝑦
  • B 1 9 6 𝑥 𝑦 , 4 2 𝑥 𝑦
  • C 1 9 6 𝑥 𝑦 , 1 4 𝑥 𝑦
  • D 5 8 8 𝑥 𝑦 , 4 2 𝑥 𝑦

Q21:

Find 𝑥 given that the ratio of the middle terms in the expansion of ( 1 + 𝑥 ) is 1 2 .

Q22:

Given that ( 1 + 𝑐 𝑥 ) = 1 + 6 𝑥 + 𝑎 𝑥 + 𝑎 𝑥 + + 𝑎 𝑥 and 2 𝑎 = 3 𝑎 , find the values of 𝑛 and 𝑐 where 𝑐 0 .

  • A 𝑛 = 3 , 𝑐 = 3
  • B 𝑛 = 4 , 𝑐 = 2
  • C 𝑛 = 4 , 𝑐 = 3
  • D 𝑛 = 3 , 𝑐 = 2

Q23:

Expand ( 5 𝑥 + 4 𝑦 ) .

  • A 6 2 5 𝑥 + 5 0 0 𝑥 𝑦 + 1 5 0 𝑥 𝑦 + 2 0 𝑥 𝑦 + 𝑦
  • B 𝑥 + 4 𝑥 𝑦 + 6 𝑥 𝑦 + 4 𝑥 𝑦 + 𝑦
  • C 𝑥 + 1 6 𝑥 𝑦 + 9 6 𝑥 𝑦 + 2 5 6 𝑥 𝑦 + 2 5 6 𝑦
  • D 6 2 5 𝑥 + 2 , 0 0 0 𝑥 𝑦 + 2 , 4 0 0 𝑥 𝑦 + 1 , 2 8 0 𝑥 𝑦 + 2 5 6 𝑦

Q24:

Expand 8 𝑥 7 4 𝑦 .

  • A 6 4 𝑥 4 𝑥 𝑦 + 𝑦 1 6
  • B 6 4 𝑥 + 2 8 𝑥 𝑦 + 4 9 𝑦 1 6
  • C 6 4 𝑥 1 1 2 𝑥 𝑦 + 4 9 𝑦
  • D 6 4 𝑥 2 8 𝑥 𝑦 + 4 9 𝑦 1 6
  • E 6 4 𝑥 + 4 𝑥 𝑦 + 𝑦 1 6

Q25:

Expand 𝑥 2 .

  • A 𝑥 + 3 2 𝑥 6 𝑥 + 2 2
  • B 𝑥 + 3 2 𝑥 + 6 𝑥 + 2 2
  • C 𝑥 3 2 𝑥 + 6 𝑥 2 2
  • D 𝑥 3 2 𝑥 + 6 𝑥 2 2

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