Worksheet: Rewriting Expressions

Practice in this worksheet rewriting expressions by expanding, factoring, and simplifying, and become familiar with some useful identities.

Q1:

Factorise fully 6 4 𝑥 8 1 2 .

  • A ( 9 𝑥 + 8 ) ( 9 𝑥 8 )
  • B ( 8 𝑥 9 ) 2
  • C ( 𝑥 + 7 2 ) ( 𝑥 7 2 )
  • D ( 8 𝑥 + 9 ) ( 8 𝑥 9 )
  • E ( 𝑥 7 2 ) 2

Q2:

Factor the expression 𝑥 4 9 2 .

  • A ( 𝑥 7 ) 2
  • B ( 2 𝑥 + 7 ) 2
  • C ( 2 𝑥 7 ) 2
  • D ( 𝑥 + 7 ) ( 𝑥 7 )
  • E ( 𝑥 + 4 9 ) ( 𝑥 4 9 )

Q3:

Factor 𝑎 6 𝑎 𝑏 + 9 𝑏 .

  • A ( 𝑎 + 𝑏 ) ( 𝑎 + 2 𝑏 )
  • B ( 𝑎 + 3 𝑏 )
  • C ( 2 𝑎 𝑏 ) ( 𝑎 2 𝑏 )
  • D ( 𝑎 3 𝑏 )
  • E ( 𝑎 + 𝑏 ) ( 𝑎 + 3 𝑏 )

Q4:

Factor 𝑥 + 6 𝑥 + 9 2 .

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

Q5:

Determine which of the following expressions is equivalent to 𝑝 𝑞 𝑝 + 𝑞 2 2 .

  • A 𝑝 2 𝑞
  • B 𝑝 + 𝑞
  • C 𝑝 + 2 𝑞
  • D 𝑝 𝑞
  • E 2 𝑝 + 2 𝑞

Q6:

An identity is an equation that is true for all values of its variables.

For example, 2 ( 𝑎 + 𝑏 ) = 2 𝑎 + 2 𝑏 is an identity because it will be true for all values of 𝑎 and 𝑏 .

Expand and simplify 𝑥 𝑦 + ( 2 𝑥 𝑦 ) 2 2 2 2 .

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

Factor 𝑥 + 2 𝑥 𝑦 + 𝑦 4 2 2 4 .

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

Is the equation 𝑥 + 𝑦 = 𝑥 𝑦 + ( 2 𝑥 𝑦 ) 2 2 2 2 2 2 2 an identity?

  • Ano
  • Byes

Substitute 𝑥 = 3 and 𝑦 = 2 into the identity ( 𝑥 + 𝑦 ) = ( 𝑥 𝑦 ) + ( 2 𝑥 𝑦 ) 2 2 2 2 2 2 2 to generate a Pythagorean triple.

  • A 1 3 = 5 + 1 2 2 2 2
  • B 1 3 = 1 2 5 2 2 2
  • C 1 3 = ( 5 + 1 2 ) 2 2
  • D 5 = ( 6 1 ) 2 2
  • E 5 = 3 + 4 2 2 2

Q7:

Answer the following questions for the brackets ( 𝑥 𝑦 ) ( 𝑥 + 𝑦 ) .

Expand the brackets ( 𝑥 𝑦 ) ( 𝑥 + 𝑦 ) .

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

Is the identity ( 𝑥 𝑦 ) ( 𝑥 + 𝑦 ) = 𝑥 𝑦 2 2 true?

  • Ayes
  • Bno

Q8:

Is the equation 𝑥 + 8 𝑥 + 1 3 = ( 𝑥 + 8 ) 8 𝑥 5 1 2 2 an identity?

  • A Yes
  • B No

Q9:

Factor the expression 4 𝑎 9 𝑏 2 2 .

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

Q10:

Factorise fully 𝑎 𝑏 ( 𝑎 𝑏 5 ) 2 2 2 .

  • A 5 ( 𝑎 𝑏 5 )
  • B 5 ( 2 𝑎 𝑏 5 )
  • C 5 ( 𝑎 𝑏 + 5 )
  • D 5 ( 2 𝑎 𝑏 5 )
  • E ( 𝑎 𝑏 + 5 ) 2

Q11:

Factorise fully 𝑏 ( 𝑎 + 8 𝑏 ) 𝑏 ( 𝑎 + 8 𝑏 ) 3 3 .

  • A 𝑏 ( 𝑎 + 8 𝑏 ) ( 𝑎 + 7 𝑏 ) 2
  • B 𝑏 ( 𝑎 + 8 𝑏 ) ( 𝑎 + 7 𝑏 ) ( 𝑎 + 9 𝑏 ) 2
  • C 𝑏 ( 𝑎 + 7 𝑏 ) ( 𝑎 + 9 𝑏 ) 2
  • D 𝑏 ( 𝑎 + 8 𝑏 ) ( 𝑎 + 7 𝑏 ) ( 𝑎 + 9 𝑏 )
  • E 𝑏 ( 𝑎 + 8 𝑏 ) ( 𝑎 + 9 𝑏 ) 2

Q12:

Factorise fully ( 5 𝑎 3 ) 3 6 2 .

  • A ( 5 𝑎 + 3 ) ( 𝑎 9 )
  • B 5 ( 𝑎 + 3 ) ( 𝑎 6 )
  • C ( 5 𝑎 9 ) 2
  • D ( 5 𝑎 + 3 ) ( 5 𝑎 9 )
  • E ( 5 𝑎 + 3 ) 2

Q13:

Expand and simplify ( 2 𝑥 3 𝑦 ) 5 𝑥 5 𝑥 𝑦 𝑦 2 2 .

  • A 1 0 𝑥 1 5 𝑥 𝑦 1 0 𝑥 2 𝑥 𝑦 + 1 5 𝑥 𝑦 + 3 𝑦 3 2 2 2 3
  • B 1 0 𝑥 2 5 𝑥 𝑦 5 𝑥 𝑦 3 𝑦 3 2 2 3
  • C 1 0 𝑥 1 0 𝑥 𝑦 + 1 3 𝑥 𝑦 1 5 𝑥 𝑦 + 3 𝑦 3 2 2 2
  • D 1 0 𝑥 2 5 𝑥 𝑦 + 1 3 𝑥 𝑦 + 3 𝑦 3 2 2 3

Q14:

Write two equivalent expressions for the area of the following figure.

  • A 1 0 ( 𝑥 + 8 ) , 𝑥 + 8 0
  • B 1 0 ( 𝑥 + 8 ) , 1 0 𝑥 + 1 8
  • C 2 0 ( 𝑥 + 8 ) , 2 0 𝑥 + 1 6 0
  • D 1 0 ( 𝑥 + 8 ) , 1 0 𝑥 + 8 0
  • E 2 0 ( 𝑥 + 8 ) , 2 0 𝑥 + 2 8

Q15:

Express the following using symbols: The product of 4 and 16 plus 11.

  • A 4 + 1 6 × 1 1
  • B 4 × 1 6 1 1
  • C 4 1 6 × 1 1
  • D 4 × 1 6 + 1 1
  • E 4 × 1 1 + 1 6

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