Worksheet: Factorization Using the Highest Common Factor

In this worksheet, we will practice factoring algebraic expressions by identifying the highest common factor (HCF).

Q1:

Write 35𝑥+910 in the form 310(𝑎𝑥+𝑏).

  • A 3 1 0 3 5 𝑥 + 3
  • B 3 1 0 ( 2 𝑥 + 3 )
  • C 3 1 0 ( 𝑥 + 3 )
  • D 3 5 𝑥 + 9 1 0
  • E 3 1 0 2 𝑥 + 9 1 0

Q2:

Factor 6𝑥+24.

  • A 6 ( 6 𝑥 + 4 )
  • B 6 ( 𝑥 + 4 )
  • C 3 ( 2 𝑥 + 4 )
  • D 3 ( 2 𝑥 + 2 4 )
  • E 6 ( 𝑥 + 2 4 )

Q3:

Factor 2015𝑥.

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

Q4:

Factor the expression 3𝑝𝑞+𝑝 completely.

  • A 𝑝 ( 3 𝑞 + 1 )
  • B 3 𝑝 ( 𝑞 + 1 )
  • C 3 𝑝 ( 𝑞 + 𝑝 )
  • D 𝑝 ( 𝑞 + 1 )
  • E 𝑝 ( 3 𝑞 + 𝑝 )

Q5:

Factor 2𝑎𝑏+8𝑎 completely.

  • A 2 ( 𝑏 + 4 𝑎 )
  • B 2 𝑎 ( 2 𝑏 + 4 )
  • C 2 𝑎 ( 𝑏 + 8 )
  • D 2 𝑎 ( 2 𝑏 + 8 )
  • E 2 𝑎 ( 𝑏 + 4 )

Q6:

Factor the expression 3𝑝𝑛+1𝑛1 completely.

  • A 𝑛 1 ( 3 𝑝 1 )
  • B 𝑛 ( 3 𝑝 1 ) + 3 𝑝 1
  • C ( 3 𝑝 1 ) 𝑛 + 1
  • D 𝑛 1 ( 3 𝑝 + 1 )
  • E 𝑛 + 1 ( 3 𝑝 + 1 )

Q7:

Factor the expression 6𝑝+3𝑝6𝑝𝑞 completely.

  • A 3 2 𝑝 2 𝑝 𝑞 + 𝑝
  • B 3 𝑝 ( 3 𝑝 3 𝑞 + 1 )
  • C 6 𝑝 ( 𝑝 𝑞 + 3 )
  • D 3 𝑝 ( 2 𝑝 2 𝑞 + 1 )
  • E 𝑝 ( 6 𝑝 6 𝑞 + 3 )

Q8:

Write 23𝑥+16 in the form 16(𝑎𝑥+𝑏).

  • A 1 6 1 4 𝑥 + 1
  • B 1 6 4 𝑥 + 1 6
  • C 1 6 1 4 𝑥 + 1 6
  • D 1 6 ( 4 𝑥 + 1 )
  • E 1 6 1 9 𝑥 + 1

Q9:

Factor 15𝑒+15𝑓 completely.

  • A 1 5 𝑒 ( 𝑒 + 1 5 𝑓 )
  • B 1 5 𝑒 ( 𝑒 + 𝑓 )
  • C 1 5 ( 𝑒 + 𝑓 )
  • D 1 5 ( 1 5 𝑒 + 𝑓 )
  • E 1 5 ( 𝑒 + 1 5 𝑓 )

Q10:

Factor 8𝑥+40𝑦 completely.

  • A 8 ( 𝑥 + 4 0 𝑦 )
  • B 4 ( 2 𝑥 + 4 0 𝑦 )
  • C 8 ( 𝑥 + 5 𝑦 )
  • D 4 ( 𝑥 + 4 0 𝑦 )
  • E 8 ( 8 𝑥 + 5 𝑦 )

Q11:

Factor 𝑥+36𝑥 completely.

  • A ( 𝑥 + 6 𝑖 ) ( 𝑥 6 𝑖 )
  • B ( 𝑥 + 6 𝑖 ) ( 𝑥 6 𝑖 )
  • C 𝑥 ( 𝑥 + 6 𝑖 ) ( 𝑥 6 𝑖 )
  • D ( 𝑥 + 6 ) ( 𝑥 6 )
  • E 𝑥 ( 𝑥 + 6 ) ( 𝑥 6 )

Q12:

Factor the expression 4𝑏𝑐+2𝑏𝑑 completely.

  • A 𝑏 ( 4 𝑐 + 2 𝑑 )
  • B 2 𝑏 ( 2 𝑐 + 𝑑 )
  • C 2 ( 2 𝑏 𝑐 + 𝑏 𝑑 )
  • D 2 𝑏 𝑐 ( 2 + 𝑑 )
  • E 2 𝑏 ( 𝑐 + 𝑑 )

Q13:

Factor the expression 2(𝑝1)+𝑥(𝑝1) completely.

  • A ( 𝑝 1 ) ( 𝑥 + 1 )
  • B ( 𝑝 1 ) ( 𝑥 + 2 )
  • C ( 𝑝 2 ) ( 𝑥 1 )
  • D ( 𝑝 1 ) ( 2 𝑥 + 2 )
  • E 𝑝 ( 𝑥 1 )

Q14:

Given that 𝑎𝑏=32, find 𝑎(𝑎𝑏)+𝑏(𝑏𝑎).

Q15:

Which of the following expressions is equivalent to 34𝑥512?

  • A 1 4 ( 3 𝑥 5 )
  • B 1 1 2 ( 9 𝑥 5 )
  • C 1 1 2 ( 9 𝑥 + 5 )
  • D 1 4 ( 3 𝑥 + 5 )
  • E 3 4 𝑥 5 1 2

Q16:

Write a completely factored expression for the total surface area of a rectangular prism whose dimensions are 𝑙, 𝑤, and .

  • A 2 ( 𝑤 + 𝑙 + )
  • B 2 𝑙 ( 𝑤 + + 𝑤 )
  • C 2 ( 𝑙 𝑤 + 𝑙 + 𝑤 )
  • D 𝑙 𝑤 + 𝑙 + 𝑤
  • E 2 ( 𝑙 𝑤 + 𝑙 + 𝑤 )

Q17:

Factor 12𝑥8.

  • A 4 ( 3 𝑥 8 )
  • B 4 ( 1 2 𝑥 2 )
  • C 4 ( 3 𝑥 2 )
  • D 2 ( 6 𝑥 8 )
  • E 2 ( 6 𝑥 2 )

Q18:

Find the highest common factor of the two terms in this expression: 4𝑥18𝑥.

  • A 4 𝑥
  • B 2 𝑥
  • C 4 𝑥
  • D 1 8 𝑥
  • E 2 𝑥

Q19:

What is the highest common factor of the expression 3𝑥𝑦11𝑥𝑦?

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

Q20:

Factorise fully 24𝑥+72𝑥+54𝑥.

  • A 𝑥 ( 1 2 𝑥 + 3 )
  • B 𝑥 ( 3 𝑥 + 2 )
  • C 𝑥 ( 2 𝑥 + 3 )
  • D 6 𝑥 ( 2 𝑥 + 3 )
  • E 6 𝑥 ( 3 𝑥 + 2 )

Q21:

Factorise fully (𝑎10)(𝑎+8)2(𝑎+8).

  • A ( 𝑎 + 3 ) ( 𝑎 3 2 )
  • B 𝑎 + 8 𝑎 1 2
  • C ( 𝑎 + 8 ) ( 𝑎 1 2 )
  • D ( 𝑎 8 ) ( 𝑎 + 1 2 )
  • E ( 𝑎 + 4 ) ( 𝑎 2 4 )

Q22:

Factorise fully 𝑥(𝑥𝑦)𝑦(𝑦𝑥) .

  • A ( 𝑦 + 𝑥 ) ( 𝑦 𝑥 )
  • B ( 𝑥 + 𝑦 ) ( 𝑥 𝑦 )
  • C ( 𝑥 𝑦 )
  • D ( 𝑥 𝑦 ) ( 𝑦 + 𝑥 )
  • E ( 𝑥 + 𝑦 )

Q23:

Factor 15+20𝑡 completely.

  • A 5 ( 1 5 + 4 𝑡 )
  • B 5 ( 3 + 2 0 𝑡 )
  • C 5 ( 3 + 4 𝑡 )
  • D 1 5 ( 1 + 2 0 𝑡 )
  • E 1 5 ( 3 + 4 𝑡 )

Q24:

Using the diagram, factor 4𝑥+12.

  • A 3 ( 𝑥 + 4 )
  • B 4 ( 4 𝑥 + 3 )
  • C 4 ( 𝑥 + 3 )
  • D 3 ( 4 𝑥 + 1 2 )
  • E 4 ( 𝑥 + 1 2 )

Q25:

Using the diagram, factor 6𝑥+9𝑦.

  • A 3 ( 2 𝑥 + 3 𝑦 )
  • B 6 ( 𝑥 + 3 𝑦 )
  • C 6 ( 𝑥 + 9 𝑦 )
  • D 3 ( 2 𝑥 + 9 𝑦 )
  • E 3 ( 6 𝑥 + 3 𝑦 )

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