Lesson Worksheet: The Binomial Theorem Mathematics

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

Q1:

Use the binomial theorem to find the expansion of (1+𝑥).

  • A1+4𝑥+6𝑥+4𝑥+𝑥
  • B1+𝑥
  • C1+4𝑥+6𝑥+4𝑥+4𝑥
  • D𝑥+4𝑥+4𝑥+𝑥
  • E1+3𝑥+6𝑥+10𝑥+15𝑥

Q2:

Expand (7+2𝑥).

  • A𝑥+21𝑥147𝑥+343
  • B8𝑥+84𝑥+294𝑥+343
  • C8𝑥+84𝑥294𝑥+343
  • D𝑥+21𝑥+147𝑥+343

Q3:

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

  • A𝑎+8𝑎𝑏+24𝑎𝑏+32𝑎𝑏+64𝑏
  • B𝑎+4𝑎𝑏+6𝑎𝑏+4𝑎𝑏+𝑏
  • C𝑎+8𝑎𝑏+24𝑎𝑏+32𝑎𝑏+16𝑏
  • D𝑎+16𝑏
  • E𝑎+4𝑎𝑏+24𝑎𝑏+32𝑎𝑏+16𝑏

Q4:

Use the binomial theorem to find the expansion of (𝑎𝑏).

  • A𝑎5𝑎𝑏+10𝑎𝑏10𝑎𝑏+5𝑎𝑏𝑏
  • B𝑎+5𝑎𝑏+10𝑎𝑏+10𝑎𝑏+5𝑎𝑏+𝑏
  • C𝑎+5𝑎𝑏10𝑎𝑏+10𝑎𝑏5𝑎𝑏+𝑏
  • D𝑎5𝑎𝑏10𝑎𝑏10𝑎𝑏5𝑎𝑏𝑏
  • E5𝑎5𝑎𝑏+10𝑎𝑏10𝑎𝑏+5𝑎𝑏𝑏

Q5:

Find the third term in the expansion of 10𝑥+23𝑥.

  • A4009𝑥
  • B4009𝑥
  • C8003𝑥
  • D8003𝑥

Q6:

Which of the following is equal to 𝐶+2×𝐶+3×𝐶++14×𝐶?

  • A2
  • B14×2
  • C13×2
  • D2
  • E14×2

Q7:

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

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

Q8:

Use the binomial theorem to expand (2𝑥3𝑦).

  • A8𝑥12𝑥𝑦+18𝑥𝑦27𝑦
  • B8𝑥+36𝑥𝑦+54𝑥𝑦+27𝑦
  • C8𝑥+36𝑥𝑦54𝑥𝑦+27𝑦
  • D8𝑥36𝑥𝑦+54𝑥𝑦27𝑦
  • E8𝑥36𝑥𝑦54𝑥𝑦27𝑦

Q9:

Expand (𝑥+2𝑦).

  • A𝑥+2𝑥𝑦+𝑦
  • B𝑥+4𝑥𝑦+4𝑦
  • C𝑥+2𝑥𝑦+𝑦
  • D𝑥+4𝑥𝑦+4𝑦

Q10:

Evaluate 3+1+31 using the binomial expansion theorem.

  • A36
  • B273
  • C12
  • D27
  • E123

Q11:

Expand 6𝑥13𝑥.

  • A36𝑥+4𝑥+19𝑥
  • B36𝑥12𝑥+1𝑥
  • C36𝑥4𝑥+19𝑥
  • D𝑥2𝑥3+19𝑥

Q12:

Expand 𝑥41𝑥.

  • A𝑥1,0245𝑥256+5𝑥3258𝑥+54𝑥1𝑥
  • B𝑥20𝑥+160𝑥640𝑥+1,280𝑥1,024
  • C𝑥20𝑥+160𝑥640𝑥+1,280𝑥1,024𝑥
  • D𝑥5𝑥+10𝑥10𝑥+5𝑥1𝑥

Q13:

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

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

  • A𝑘=154
  • B𝑘=12
  • C𝑘=14
  • D𝑘=2
  • E𝑘=1

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

  • A12
  • B3256
  • C15
  • D384
  • E38

Q14:

Given that (1+𝑐𝑥)=1+6𝑥+𝑎𝑥+𝑎𝑥++𝑎𝑥 and 2𝑎=3𝑎, find the values of 𝑛 and 𝑐 where 𝑐0.

  • A𝑛=4, 𝑐=3
  • B𝑛=3, 𝑐=3
  • C𝑛=3, 𝑐=2
  • D𝑛=4, 𝑐=2

Q15:

Expand (5𝑥+4𝑦).

  • A𝑥+4𝑥𝑦+6𝑥𝑦+4𝑥𝑦+𝑦
  • B625𝑥+2,000𝑥𝑦+2,400𝑥𝑦+1,280𝑥𝑦+256𝑦
  • C625𝑥+500𝑥𝑦+150𝑥𝑦+20𝑥𝑦+𝑦
  • D𝑥+16𝑥𝑦+96𝑥𝑦+256𝑥𝑦+256𝑦

Q16:

Expand 𝑥2.

  • A𝑥+32𝑥6𝑥+22
  • B𝑥32𝑥+6𝑥22
  • C𝑥+32𝑥+6𝑥+22
  • D𝑥32𝑥+6𝑥22

Q17:

Find the third term in the expansion of (4𝑥+3).

  • A27𝑥
  • B27𝑥
  • C108𝑥
  • D108𝑥

Q18:

Find the value of 𝑥 that satisfies 1+9𝑥+9×82×1𝑥+9×8×73×2×1𝑥++𝑥=512.

Q19:

Use the binomial theorem to find the expansion of 𝑎𝑏2.

  • A𝑎3𝑎𝑏23𝑎𝑏4𝑏8
  • B𝑎3𝑎𝑏2+3𝑎𝑏4+𝑏8
  • C𝑎+3𝑎𝑏2+3𝑎𝑏4+𝑏8
  • D𝑎3𝑎𝑏8+3𝑎𝑏8𝑏8
  • E𝑎3𝑎𝑏2+3𝑎𝑏4𝑏8

Q20:

Use the binomial theorem to expand 𝑥2+𝑦3.

  • A𝑥16+𝑥𝑦6+𝑥𝑦6+2𝑥𝑦9+𝑦81
  • B𝑥16+𝑥𝑦2+𝑥𝑦6+2𝑥𝑦27+𝑦81
  • C𝑥16+𝑥𝑦6+𝑥𝑦36+2𝑥𝑦27+𝑦81
  • D𝑥16+𝑥𝑦6+𝑥𝑦6+2𝑥𝑦27+𝑦81
  • E𝑥16+𝑥𝑦24+𝑥𝑦36+𝑥𝑦54+𝑦81

Q21:

Use the binomial theorem to expand 2𝑥34𝑦5.

  • A16𝑥81+128𝑥𝑦135128𝑥𝑦75+512𝑥𝑦375256𝑦625
  • B16𝑥81128𝑥𝑦135+128𝑥𝑦75128𝑥𝑦375+256𝑦625
  • C16𝑥81128𝑥𝑦135+128𝑥𝑦75+512𝑥𝑦375+256𝑦625
  • D16𝑥81128𝑥𝑦135+128𝑥𝑦75512𝑥𝑦375+256𝑦625
  • E16𝑥81+128𝑥𝑦135+128𝑥𝑦75+512𝑥𝑦375+256𝑦625

Q22:

Expand and simplify 9+𝑥+9𝑥.

  • A2𝑥+162
  • B36𝑥+2𝑥+162
  • C2𝑥+162
  • D36𝑥
  • E36𝑥

Q23:

Expand (4𝑥)(4+𝑥).

  • A𝑥+8𝑥+16
  • B𝑥8𝑥+16
  • C𝑥+32𝑥+256
  • D𝑥32𝑥+256

Q24:

Expand 10+𝑥+108.

  • A𝑥+30𝑥+300𝑥+1,008
  • B𝑥300𝑥30𝑥6162+240
  • C𝑥+300𝑥+30𝑥6162+2,240
  • D𝑥30𝑥300𝑥992
  • E60𝑥+2,000

Q25:

Using the binomial theorem, expand 𝑥2𝑥+1𝑥+2𝑥+1+2𝑥.

  • A𝑥+4𝑥+6𝑥+4𝑥+1
  • B𝑥4𝑥+6𝑥4𝑥+1
  • C𝑥+4𝑥+6𝑥+4𝑥+1
  • D𝑥4𝑥+6𝑥4𝑥+1

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