Worksheet: Simplifying Multivariable Polynomial Expressions

In this worksheet, we will practice simplifying multivariable polynomials by expanding parentheses, by multiplying by monomials, and by adding and subtracting terms of the same power.

Q1:

Simplify 2 𝑥 4 𝑥 9 5 𝑥 3 𝑥 + 1 3 2 3 .

  • A 3 𝑥 4 𝑥 + 3 𝑥 + 1 0 3 2
  • B 3 𝑥 4 𝑥 + 3 𝑥 1 0 3 2
  • C 3 𝑥 4 𝑥 3 𝑥 1 0 3 2
  • D 3 𝑥 4 𝑥 + 3 𝑥 1 0 3 2
  • E 3 𝑥 4 𝑥 3 𝑥 8 3 2

Q2:

Simplify 6 ( 3 𝑏 + 2 ) + 4 ( 2 𝑏 + 4 ) .

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

Q3:

Simplify 6 5 𝑧 + 2 2 ( 𝑧 4 ) 2 .

  • A 3 0 𝑧 2 𝑧 2 0 2
  • B 3 0 𝑧 + 2 𝑧 + 2 0 2
  • C 3 2 𝑧 + 2 0 2
  • D 3 0 𝑧 2 𝑧 + 2 0 2
  • E 3 0 𝑧 2 𝑧 + 4 2

Q4:

Simplify 6 𝑥 ( 𝑥 + 𝑦 ) 𝑦 ( 6 𝑥 𝑦 ) + 6 ( 𝑦 𝑥 ) 6 6 7 7 .

  • A 5 𝑦 + 1 2 𝑥 7 7
  • B 5 𝑦 7
  • C 7 𝑦 1 2 𝑥 𝑦 7
  • D 7 𝑦 7

Q5:

Express, in terms of 𝑥 , 𝑦 , and 𝑧 , the sum of the surface areas of the two given figures.

  • A 2 4 4 𝑥 𝑦 + 1 6 8 𝑥 𝑧 + 1 4 0 𝑦 𝑧
  • B 4 0 0 𝑥 𝑦 + 2 5 9 𝑥 𝑧 + 2 2 4 𝑦 𝑧
  • C 3 3 2 𝑥 𝑦 + 2 4 5 𝑥 𝑧 + 1 9 6 𝑦 𝑧
  • D 4 8 8 𝑥 𝑦 + 3 3 6 𝑥 𝑧 + 2 8 0 𝑦 𝑧
  • E 5 7 6 𝑥 𝑦 + 4 1 3 𝑥 𝑧 + 3 3 6 𝑦 𝑧

Q6:

Find an expression for the area of the shape below.

  • A 2 4 1 𝑥 2
  • B 2 3 9 𝑥
  • C 2 4 1 𝑥
  • D 2 3 9 𝑥 2
  • E 2 4 0 𝑥 2

Q7:

Expand and simplify 7 𝑥 ( 𝑥 + 𝑦 ) + 𝑦 [ 3 𝑥 4 𝑦 ( 𝑥 + 2 𝑦 ) ] + 7 𝑥 [ 3 𝑥 2 ( 𝑥 + 3 𝑦 ) ] .

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

Q8:

Is the equation 𝑥 𝑦 𝑥 + 𝑦 = 𝑥 𝑦 4 4 2 2 2 2 an identity?

  • A yes
  • Bno

Q9:

Is the equation 𝑥 + 𝑦 𝑥 𝑦 = 𝑥 + 𝑦 4 4 2 2 2 2 an identity?

  • A no
  • B yes

Q10:

Factorise fully 𝑎 1 4 𝑎 𝑏 + 4 8 𝑏 4 2 2 4 .

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

Q11:

Which of the following is equivalent to ( 𝑥 𝑦 ) ( 𝑥 + 𝑦 ) 𝑥 2 𝑥 𝑦 + 𝑦 4 2 2 4 ?

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

Q12:

Simplify 8 𝑥 × 3 𝑦 × 5 𝑧 .

  • A 4 0 𝑥 𝑦 𝑧
  • B 1 6 𝑥 𝑦 𝑧
  • C 1 2 0 ( 𝑥 + 𝑦 + 𝑧 )
  • D 1 2 0 𝑥 𝑦 𝑧
  • E 1 6 ( 𝑥 + 𝑦 + 𝑧 )

Q13:

Simplify 3 𝑥 [ 𝑥 + ( 𝑦 𝑥 ) ] + 𝑦 [ 2 𝑦 + 2 ( 𝑥 𝑦 ) ] , and find the value of the expression when 𝑥 = 𝑦 = 1 .

  • A 5 𝑥 𝑦 2 𝑦 2 , 3
  • B 3 𝑥 + 3 𝑥 𝑦 + 3 𝑥 + 2 𝑥 𝑦 2 𝑦 + 2 𝑦 3 2 2 2 3 2 , 5
  • C 5 𝑥 𝑦 , 3
  • D 5 𝑥 𝑦 , 5

Q14:

Determine which of the following expressions is equivalent to ( 𝑥 + 𝑦 ) 3 .

  • A 𝑥 + 𝑦 3 3
  • B 𝑥 3 𝑥 𝑦 3 𝑥 𝑦 + 𝑦 3 2 2 3
  • C 𝑥 + 3 𝑥 𝑦 + 3 𝑥 𝑦 𝑦 3 2 2 3
  • D 𝑥 + 3 𝑥 𝑦 + 3 𝑥 𝑦 + 𝑦 3 2 2 3
  • E 𝑥 + 3 𝑥 𝑦 + 3 𝑥 𝑦 + 𝑦 3 2 2 2 3

Q15:

Express, in terms of 𝑥 , 𝑦 , and 𝑧 , the sum of the surface areas of the two given figures.

  • A 3 0 2 𝑥 𝑦 + 1 5 9 𝑥 𝑧 + 1 1 8 𝑦 𝑧
  • B 4 2 2 𝑥 𝑦 + 2 3 4 𝑥 𝑧 + 1 5 8 𝑦 𝑧
  • C 4 8 4 𝑥 𝑦 + 2 4 3 𝑥 𝑧 + 1 9 6 𝑦 𝑧
  • D 6 0 4 𝑥 𝑦 + 3 1 8 𝑥 𝑧 + 2 3 6 𝑦 𝑧
  • E 7 8 6 𝑥 𝑦 + 4 0 2 𝑥 𝑧 + 3 1 4 𝑦 𝑧

Q16:

Simplify 8 𝑥 ( 𝑥 + 𝑦 ) 𝑦 ( 8 𝑥 𝑦 ) + 8 ( 𝑦 𝑥 ) 7 7 8 8 .

  • A 7 𝑦 + 1 6 𝑥 8 8
  • B 7 𝑦 8
  • C 9 𝑦 1 6 𝑥 𝑦 8
  • D 9 𝑦 8

Q17:

Simplify 8 𝑥 ( 𝑥 + 𝑦 ) 𝑦 ( 8 𝑥 𝑦 ) 8 ( 𝑦 𝑥 ) 7 7 8 8 .

  • A 9 𝑦 1 6 𝑥 8 8
  • B 9 𝑦 8
  • C 7 𝑦 + 1 6 𝑥 𝑦 8
  • D 7 𝑦 8

Q18:

Expand and simplify 4 𝑥 ( 𝑥 6 𝑦 ) + 6 𝑦 [ 4 𝑥 + 6 𝑦 ( 𝑥 + 𝑦 ) ] 2 𝑥 [ 𝑥 + 5 ( 𝑥 2 𝑦 ) ] .

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

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