Worksheet: Simplifying Algebraic Fractions

In this worksheet, we will practice adding and simplifying algebraic fractions containing linear expressions with 2 or more variables.

Q1:

Fully simplify ๐‘ฅ โˆ’ 4 ๐‘ฅ โˆ’ 4 ๐‘ฅ + 4 2 2 .

  • A ๐‘ฅ + 2 ๐‘ฅ โˆ’ 4
  • B ๐‘ฅ โˆ’ 2 ๐‘ฅ + 2
  • C ๐‘ฅ โˆ’ 2 ๐‘ฅ โˆ’ 4
  • D ๐‘ฅ + 2 ๐‘ฅ โˆ’ 2
  • E ๐‘ฅ + 4 ๐‘ฅ + 2

Q2:

Fully simplify ( ๐‘ฅ โˆ’ 2 ) ( ๐‘ฅ โˆ’ 2 ) ( ๐‘ฅ + 2 ) 2 .

  • A ๐‘ฅ ๐‘ฅ โˆ’ 2
  • B ๐‘ฅ + 2 ๐‘ฅ โˆ’ 2
  • C ๐‘ฅ + 2 ๐‘ฅ
  • D ๐‘ฅ โˆ’ 2 ๐‘ฅ + 2
  • E 1 ๐‘ฅ + 2

Q3:

Which of the following functions are equal?

  • A ๐‘› ( ๐‘ฅ ) = 1 ๐‘ฅ ๏Šง , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 5 3 ๐‘ฅ + 5 3 ๐‘ฅ ๏Šจ ๏Šจ
  • B ๐‘› ( ๐‘ฅ ) = 1 ๐‘ฅ ๏Šง , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 5 3 ๐‘ฅ โˆ’ 5 3 ๐‘ฅ ๏Šจ ๏Šจ ๏Šฉ
  • C ๐‘› ( ๐‘ฅ ) = 1 ๐‘ฅ ๏Šง , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 5 3 ๐‘ฅ โˆ’ 5 3 ๐‘ฅ ๏Šจ ๏Šจ
  • D ๐‘› ( ๐‘ฅ ) = 1 ๐‘ฅ ๏Šง , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 5 3 ๐‘ฅ + 5 3 ๐‘ฅ ๏Šจ ๏Šจ ๏Šฉ
  • E ๐‘› ( ๐‘ฅ ) = 1 ๐‘ฅ ๏Šง , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 5 3 ๐‘ฅ ๐‘ฅ ๏Šจ ๏Šจ ๏Šจ

Q4:

Which of the following functions are equal?

  • A ๐‘› ( ๐‘ฅ ) = 9 ๐‘ฅ 9 ๐‘ฅ โˆ’ 3 6 ๏Šง , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 4 ๐‘ฅ ๐‘ฅ โˆ’ 1 6 ๏Šจ ๏Šจ ๏Šจ
  • B ๐‘› ( ๐‘ฅ ) = 9 ๐‘ฅ 9 ๐‘ฅ + 3 6 ๏Šง , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 4 ๐‘ฅ ๐‘ฅ โˆ’ 1 6 ๏Šจ ๏Šจ ๏Šจ
  • C ๐‘› ( ๐‘ฅ ) = 9 ๐‘ฅ 9 ๐‘ฅ + 3 6 ๏Šง , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ ๐‘ฅ + 8 ๐‘ฅ + 1 6 ๏Šจ ๏Šจ
  • D ๐‘› ( ๐‘ฅ ) = 9 ๐‘ฅ 9 ๐‘ฅ + 3 6 ๏Šง , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 4 ๐‘ฅ ๐‘ฅ + 8 ๐‘ฅ + 1 6 ๏Šจ ๏Šจ ๏Šจ
  • E ๐‘› ( ๐‘ฅ ) = 9 ๐‘ฅ ๐‘ฅ + 4 ๏Šง , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 4 ๐‘ฅ ๐‘ฅ + 8 ๐‘ฅ + 1 6 ๏Šจ ๏Šจ ๏Šจ

Q5:

Fully simplify ( ๐‘ฅ โˆ’ 3 ) ๏€น ๐‘ฅ โˆ’ 6 ๐‘ฅ + 9 ๏… ( ๐‘ฅ โˆ’ 3 ) 2 3 .

Q6:

Which of the following functions are equal?

  • A ๐‘› ( ๐‘ฅ ) = 8 ๐‘ฅ + 5 6 ๐‘ฅ ( ๐‘ฅ โˆ’ 3 ) ( ๐‘ฅ + 7 ) ๏Šง ๏Šฉ ๏Šจ , ๐‘› ( ๐‘ฅ ) = ๐‘ฅ ๐‘ฅ โˆ’ 3 ๏Šจ
  • B ๐‘› ( ๐‘ฅ ) = 8 ๐‘ฅ โˆ’ 5 6 ๐‘ฅ ( ๐‘ฅ โˆ’ 3 ) ( ๐‘ฅ โˆ’ 7 ) ๏Šง ๏Šฉ ๏Šจ , ๐‘› ( ๐‘ฅ ) = 8 ๐‘ฅ ๐‘ฅ โˆ’ 3 ๏Šจ
  • C ๐‘› ( ๐‘ฅ ) = 8 ๐‘ฅ + 5 6 ๐‘ฅ ( ๐‘ฅ โˆ’ 3 ) ( ๐‘ฅ + 7 ) ๏Šง ๏Šจ , ๐‘› ( ๐‘ฅ ) = 8 ๐‘ฅ ๐‘ฅ โˆ’ 3 ๏Šจ
  • D ๐‘› ( ๐‘ฅ ) = 8 ๐‘ฅ + 5 6 ๐‘ฅ ( ๐‘ฅ โˆ’ 3 ) ( ๐‘ฅ + 7 ) ๏Šง ๏Šฉ ๏Šจ , ๐‘› ( ๐‘ฅ ) = 8 ๐‘ฅ ๐‘ฅ โˆ’ 3 ๏Šจ
  • E ๐‘› ( ๐‘ฅ ) = 8 ๐‘ฅ โˆ’ 5 6 ๐‘ฅ ( ๐‘ฅ โˆ’ 3 ) ( ๐‘ฅ โˆ’ 7 ) ๏Šง ๏Šจ , ๐‘› ( ๐‘ฅ ) = 8 ๐‘ฅ ๐‘ฅ โˆ’ 3 ๏Šจ

Q7:

Simplify the function ๐‘› ( ๐‘ฅ ) = ๐‘ฅ ๐‘ฅ + 9 ๐‘ฅ + ๐‘ฅ + 9 ๐‘ฅ โˆ’ 8 1 ๏Šจ ๏Šจ , and determine its domain.

  • A ๐‘› ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 9 , domain = โ„ โˆ’ { โˆ’ 9 , 0 , 9 }
  • B ๐‘› ( ๐‘ฅ ) = 2 ๐‘ฅ ( ๐‘ฅ + 9 ) ( ๐‘ฅ โˆ’ 9 ) , domain = โ„ โˆ’ { โˆ’ 9 , 9 }
  • C ๐‘› ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 9 , domain = โ„ โˆ’ { 9 }
  • D ๐‘› ( ๐‘ฅ ) = 2 ๐‘ฅ ( ๐‘ฅ + 9 ) ( ๐‘ฅ โˆ’ 9 ) , domain = โ„ โˆ’ { โˆ’ 9 , 0 , 9 }
  • E ๐‘› ( ๐‘ฅ ) = 2 ๐‘ฅ + 9 ๐‘ฅ ( ๐‘ฅ + 9 ) ( ๐‘ฅ โˆ’ 9 ) , domain = โ„ โˆ’ { โˆ’ 9 , 0 , 9 }

Q8:

Given that ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 9 ๐‘ฅ โˆ’ 6 ๏Šง , ๐‘› ( ๐‘ฅ ) = 9 ๐‘ฅ + 8 1 ๐‘ฅ โˆ’ 6 ๏Šจ , and ๐‘› ( ๐‘ฅ ) = ๐‘› ( ๐‘ฅ ) รท ๐‘› ( ๐‘ฅ ) ๏Šง ๏Šจ , identify the domain of ๐‘› ( ๐‘ฅ ) .

  • A โ„ โˆ’ { 6 }
  • B โ„ โˆ’ { 0 , 6 }
  • C โ„ โˆ’ { โˆ’ 6 }
  • D โ„ โˆ’ { 3 , 6 }
  • E โ„ โˆ’ { โˆ’ 6 , โˆ’ 3 }

Q9:

Simplify โˆ’ 2 0 ๐‘ฅ โˆ’ 1 0 ๐‘ฅ .

Q10:

Simplify the function ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 3 4 3 7 ๐‘ฅ โˆ’ 4 9 ๐‘ฅ + 3 4 3 ๐‘ฅ ๏Šฉ ๏Šฉ ๏Šจ and find its domain.

  • A ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 7 7 ๐‘ฅ , domain = โ„ โˆ’ { 0 }
  • B ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 7 7 ๐‘ฅ , domain = โ„ โˆ’ { 0 , โˆ’ 7 }
  • C ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 7 7 ๐‘ฅ , domain = โ„ โˆ’ { 0 , 7 }
  • D ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 7 7 ๐‘ฅ , domain = โ„ โˆ’ { 0 }
  • E ๐‘“ ( ๐‘ฅ ) = ( ๐‘ฅ + 7 ) ( ๐‘ฅ + 7 ๐‘ฅ + 4 9 ) 7 ๐‘ฅ ( ๐‘ฅ โˆ’ 7 ๐‘ฅ + 4 9 ) ๏Šจ ๏Šจ , domain = โ„ โˆ’ { 0 }

Q11:

Simplify ๐‘ฅ ๐‘ฆ ๐‘ฅ ๐‘ฆ ( ๏Š ๏Šฐ ๏Šฏ ) ( ๏Š ๏Šฐ ๏Šช ) ( ๏Š ๏Šฐ ๏Šฉ ) ๏Š .

  • A ๐‘ฅ ๐‘ฆ ๏Š ๏Šฐ ๏Šฌ ๏Šช
  • B ๐‘ฅ ๐‘ฆ ๏Šง ๏Šจ ๏Šช
  • C ๐‘ฅ ๐‘ฆ ๏Š ๏Šฐ ๏Šฌ ๏Š ๏Šฐ ๏Šฉ
  • D ๐‘ฅ ๐‘ฆ ๏Šฌ ๏Šช
  • E ( ๐‘ฅ ๐‘ฆ ) ๏Šง ๏Šฆ

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