Worksheet: Multiplying and Dividing Rational Functions

In this worksheet, we will practice multiplying and dividing rational functions.

Q1:

Answer the following questions for the rational expressions 5๐‘ฅโˆ’45๐‘ฅ12๐‘ฅโˆ’4๐‘ฅ๏Šฉ๏Šจ and 15๐‘ฅโˆ’453๐‘ฅ๏Šจ.

Evaluate 5๐‘ฅโˆ’45๐‘ฅ12๐‘ฅโˆ’4๐‘ฅ๏Šฉ๏Šจ divided by 15๐‘ฅโˆ’453๐‘ฅ๏Šจ.

  • A ๐‘ฅ ( ๐‘ฅ + 3 ) 3 ๐‘ฅ โˆ’ 1 ๏Šจ
  • B ๐‘ฅ ( ๐‘ฅ + 3 ) ๐‘ฅ + 1 ๏Šจ
  • C ๐‘ฅ ( ๐‘ฅ + 3 ) 3 ๐‘ฅ โˆ’ 2 ๏Šจ
  • D ๐‘ฅ ( ๐‘ฅ + 3 ) 4 ( 3 ๐‘ฅ โˆ’ 1 ) ๏Šจ
  • E 2 5 ( ๐‘ฅ โˆ’ 3 ) ( ๐‘ฅ + 3 ) 4 ๐‘ฅ ( 3 ๐‘ฅ โˆ’ 1 ) ๏Šจ ๏Šจ

Is the result of 5๐‘ฅโˆ’45๐‘ฅ12๐‘ฅโˆ’4๐‘ฅ๏Šฉ๏Šจ divided by 15๐‘ฅโˆ’453๐‘ฅ๏Šจ a rational expression?

  • Ayes
  • Bno

Would this be true for any rational expression divided by another rational expression?

  • Ayes
  • Bno

Q2:

Answer the following questions for the rational expressions 6(๐‘ฅโˆ’2)3๐‘ฅโˆ’6๐‘ฅ๏Šจ and 6๐‘ฅโˆ’32๐‘ฅ.

Find the product of 6(๐‘ฅโˆ’2)3๐‘ฅโˆ’6๐‘ฅ๏Šจ and 6๐‘ฅโˆ’32๐‘ฅ.

  • A 3 ( 2 ๐‘ฅ โˆ’ 1 ) ๐‘ฅ ๏Šจ
  • B 3 ( 3 ๐‘ฅ โˆ’ 1 ) ๐‘ฅ ๏Šจ
  • C ๐‘ฅ + 3 2 ๐‘ฅ ๏Šจ
  • D 3 ( ๐‘ฅ โˆ’ 1 ) ๐‘ฅ ๏Šจ
  • E 2 ๐‘ฅ + 3 2 ๐‘ฅ

Is the product of 6(๐‘ฅโˆ’2)3๐‘ฅโˆ’6๐‘ฅ๏Šจ and 6๐‘ฅโˆ’32๐‘ฅ a rational expression?

  • Ano
  • Byes

Would this be true for the product of any two rational expressions?

  • Ayes
  • Bno

Q3:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅ+5๐‘ฅ+9๐‘ฅ+20ร—๐‘ฅ+15๐‘ฅ+547๐‘ฅ+69๐‘ฅ+54๏Šจ๏Šจ๏Šจ, and determine its domain.

  • A ๐‘› ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 6 ( ๐‘ฅ + 4 ) ( 7 ๐‘ฅ + 6 ) , domain =โ„โˆ’๏ฌโˆ’4,โˆ’67๏ธ
  • B ๐‘› ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 6 ( ๐‘ฅ โˆ’ 4 ) ( 7 ๐‘ฅ โˆ’ 6 ) , domain =โ„โˆ’๏ฌโˆ’9,โˆ’5,โˆ’4,โˆ’67๏ธ
  • C ๐‘› ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 6 ( ๐‘ฅ + 4 ) ( 7 ๐‘ฅ + 6 ) , domain =โ„โˆ’๏ฌโˆ’9,โˆ’5,โˆ’4,โˆ’67๏ธ
  • D ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 6 ( ๐‘ฅ + 4 ) ( 7 ๐‘ฅ + 6 ) , domain =โ„โˆ’๏ฌโˆ’4,โˆ’67๏ธ
  • E ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 6 ( ๐‘ฅ + 4 ) ( 7 ๐‘ฅ + 6 ) , domain =โ„โˆ’๏ฌโˆ’9,โˆ’5,โˆ’4,โˆ’67๏ธ

Q4:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅโˆ’162๐‘ฅ+9๐‘ฅรท9๐‘ฅโˆ’72๐‘ฅ+1444๐‘ฅโˆ’81๏Šจ๏Šจ๏Šจ๏Šจ.

  • A ๐‘› ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 4 9 ๐‘ฅ ( ๐‘ฅ + 4 ) ( 2 ๐‘ฅ + 9 )
  • B ๐‘› ( ๐‘ฅ ) = ( ๐‘ฅ โˆ’ 4 ) ( 2 ๐‘ฅ + 9 ) 9 ๐‘ฅ ( ๐‘ฅ + 4 )
  • C ๐‘› ( ๐‘ฅ ) = ( ๐‘ฅ + 4 ) ( 2 ๐‘ฅ โˆ’ 9 ) 9 ๐‘ฅ ( ๐‘ฅ โˆ’ 4 )
  • D ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 4 9 ๐‘ฅ ( ๐‘ฅ โˆ’ 4 ) ( 2 ๐‘ฅ โˆ’ 9 )
  • E ๐‘› ( ๐‘ฅ ) = ( ๐‘ฅ + 4 ) ( 2 ๐‘ฅ โˆ’ 9 ) ๐‘ฅ ( ๐‘ฅ โˆ’ 4 )

Q5:

Given the function ๐‘›(๐‘ฅ)=๐‘ฅโˆ’6๐‘ฅโˆ’15๐‘ฅ+54ร—๐‘ฅโˆ’3๐‘ฅโˆ’282๐‘ฅโˆ’15๐‘ฅ+7๏Šจ๏Šจ๏Šจ, evaluate ๐‘›(7), if possible.

  • A โˆ’ 1 2
  • Bundefined
  • C โˆ’ 1 8 8
  • D โˆ’ 2

Q6:

Simplify 6๐‘ฅโˆ’3๐‘ฅ3๐‘ฅโˆ’2ร—7๐‘ฅโˆ’142๐‘ฅโˆ’1๏Šฉ๏Šจ.

  • A 4 2 ๐‘ฅ โˆ’ 1 0 5 ๐‘ฅ + 4 2 ๐‘ฅ 6 ๐‘ฅ โˆ’ 7 ๐‘ฅ + 2 ๏Šช ๏Šฉ ๏Šจ ๏Šจ
  • B 3 ๐‘ฅ ( ๐‘ฅ โˆ’ 2 ) 3 ๐‘ฅ โˆ’ 4 ๏Šจ
  • C 3 ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 1 ) 7 ( 3 ๐‘ฅ โˆ’ 2 ) ( ๐‘ฅ โˆ’ 2 ) ๏Šจ ๏Šจ
  • D 7 ๐‘ฅ ( ๐‘ฅ โˆ’ 2 ) 3 ๐‘ฅ โˆ’ 2 ๏Šจ
  • E 2 1 ๐‘ฅ ( ๐‘ฅ โˆ’ 2 ) 3 ๐‘ฅ โˆ’ 2 ๏Šจ

Q7:

Simplify 4๐‘ฅโˆ’3๐‘ฅ2๐‘ฅโˆ’1โ‹…2๐‘ฅโˆ’54๐‘ฅโˆ’2๏Šจ.

  • A ๐‘ฅ ( 4 ๐‘ฅ โˆ’ 3 ) ( 2 ๐‘ฅ โˆ’ 5 ) 2 ( 2 ๐‘ฅ โˆ’ 1 )
  • B ๐‘ฅ ( 4 ๐‘ฅ โˆ’ 5 ) ( 2 ๐‘ฅ โˆ’ 3 ) 2 ( 2 ๐‘ฅ โˆ’ 1 )
  • C ๐‘ฅ ( 4 ๐‘ฅ โˆ’ 3 ) ( 2 ๐‘ฅ โˆ’ 5 ) 2 ( 2 ๐‘ฅ โˆ’ 1 ) ๏Šจ
  • D 8 ๐‘ฅ โˆ’ 2 6 ๐‘ฅ + 1 5 ๐‘ฅ 8 ๐‘ฅ โˆ’ 8 ๐‘ฅ + 2 ๏Šฉ ๏Šจ ๏Šจ
  • E ( 4 ๐‘ฅ โˆ’ 3 ) ( 2 ๐‘ฅ โˆ’ 5 ) 2 ( 2 ๐‘ฅ โˆ’ 1 ) ๏Šจ

Q8:

Simplify 14๐‘ฅโˆ’21๐‘ฅ4๐‘ฅโˆ’20รท4๐‘ฅโˆ’62๐‘ฅโˆ’1๏Šจ.

  • A 1 4 ๐‘ฅ โˆ’ 3 ๐‘ฅ 8 ๐‘ฅ + 4 0 ๏Šจ
  • B 7 ๐‘ฅ โˆ’ 3 ๐‘ฅ 8 ๐‘ฅ โˆ’ 2 0 ๏Šจ
  • C 7 ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 1 ) 8 ( ๐‘ฅ + 5 )
  • D 7 ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 3 ) 2 ( ๐‘ฅ โˆ’ 5 ) ( 2 ๐‘ฅ โˆ’ 1 ) ๏Šจ
  • E 7 ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 1 ) 8 ( ๐‘ฅ โˆ’ 5 )

Q9:

Determine the domain of the function ๐‘›(๐‘ฅ)=3๐‘ฅโˆ’15๐‘ฅโˆ’6รท6๐‘ฅโˆ’304๐‘ฅโˆ’24.

  • A โ„ โˆ’ { โˆ’ 6 , โˆ’ 5 }
  • B โ„ โˆ’ { 5 , 6 }
  • C โ„ โˆ’ { 6 }
  • D โ„ โˆ’ { 5 }
  • E โ„

Q10:

Find the volume of a cube whose side length is 45๐‘ฅ.

  • A 1 6 2 5 ๐‘ฅ ๏Šจ
  • B 6 4 1 2 5 ๐‘ฅ ๏Šฉ
  • C 6 4 1 2 5
  • D 6 4 1 2 5 ๐‘ฅ
  • E 4 5 ๐‘ฅ ๏Šฉ

Q11:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅ+16๐‘ฅ+64๐‘ฅ+8๐‘ฅร—7๐‘ฅโˆ’5664โˆ’๐‘ฅ๏Šจ๏Šจ๏Šจ, and determine its domain.

  • A ๐‘› ( ๐‘ฅ ) = โˆ’ 7 ๐‘ฅ , domain =โ„โˆ’{0}
  • B ๐‘› ( ๐‘ฅ ) = โˆ’ 7 ๐‘ฅ , domain =โ„โˆ’{โˆ’8,0,8}
  • C ๐‘› ( ๐‘ฅ ) = โˆ’ 1 7 ๐‘ฅ , domain =โ„โˆ’{โˆ’8,0,8}
  • D ๐‘› ( ๐‘ฅ ) = 7 ๐‘ฅ , domain =โ„โˆ’{โˆ’8,0,8}
  • E ๐‘› ( ๐‘ฅ ) = 7 ๐‘ฅ , domain =โ„โˆ’{0}

Q12:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅ+3432๐‘ฅ+14๐‘ฅร—๐‘ฅ+3๐‘ฅโˆ’7๐‘ฅ+49๏Šฉ๏Šจ๏Šจ, and determine its domain.

  • A ๐‘› ( ๐‘ฅ ) = 2 ๐‘ฅ ๐‘ฅ + 3 , domain =โ„โˆ’{โˆ’7,0}
  • B ๐‘› ( ๐‘ฅ ) = 2 ๐‘ฅ ๐‘ฅ + 3 , domain =โ„โˆ’{0}
  • C ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 3 2 ๐‘ฅ , domain =โ„โˆ’{0}
  • D ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 3 2 ๐‘ฅ , domain =โ„โˆ’{โˆ’7,0}
  • E ๐‘› ( ๐‘ฅ ) = ๐‘ฅ 2 ( ๐‘ฅ + 3 ) , domain =โ„โˆ’{โˆ’7,0}

Q13:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅโˆ’12๐‘ฅ+36๐‘ฅโˆ’216รท7๐‘ฅโˆ’42๐‘ฅ+6๐‘ฅ+36๏Šจ๏Šฉ๏Šจ, and determine its domain.

  • A ๐‘› ( ๐‘ฅ ) = 1 7 , domain =โ„
  • B ๐‘› ( ๐‘ฅ ) = 1 7 , domain =โ„โˆ’{6}
  • C ๐‘› ( ๐‘ฅ ) = 7 , domain =โ„โˆ’{6}
  • D ๐‘› ( ๐‘ฅ ) = 1 6 , domain =โ„โˆ’{6}
  • E ๐‘› ( ๐‘ฅ ) = 7 , domain =โ„

Q14:

Determine the domain of the function ๐‘›(๐‘ฅ)=๐‘ฅโˆ’๐‘ฅโˆ’6๐‘ฅโˆ’4รท2๐‘ฅโˆ’6๐‘ฅโˆ’4๐‘ฅ+4๏Šจ๏Šจ๏Šจ.

  • A โ„ โˆ’ { โˆ’ 3 , โˆ’ 2 }
  • B โ„
  • C โ„ โˆ’ { โˆ’ 2 , 2 }
  • D โ„ โˆ’ { โˆ’ 2 , 2 , 3 }
  • E โ„ โˆ’ { โˆ’ 3 , โˆ’ 2 , 2 }

Q15:

Find the missing term in the equality 23ร—๏€ผโˆ’1+18๏ˆ=23ร—(โˆ’1)+23ร—.

  • A 3 2
  • B 1 8
  • C โˆ’ 1 6
  • D โˆ’ 1

Q16:

Given that ๐‘›(๐‘ฅ)=๐‘ฅ+9๐‘ฅโˆ’6๏Šง, ๐‘›(๐‘ฅ)=9๐‘ฅ+81๐‘ฅโˆ’6๏Šจ, and ๐‘›(๐‘ฅ)=๐‘›(๐‘ฅ)รท๐‘›(๐‘ฅ)๏Šง๏Šจ, identify the domain of ๐‘›(๐‘ฅ).

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

Q17:

Given that ๐‘“(๐‘ฅ)=๐‘ฅ+9๐‘ฅ+14๐‘ฅโˆ’4รท๐‘ฅโˆ’49๐‘ฅโˆ’2๐‘ฅ๏Šจ๏Šจ๏Šจ๏Šจ and ๐‘“(๐‘Ž)=4, find the value of ๐‘Ž.

  • A โˆ’ 2 8 3
  • B 2 8 3
  • C โˆ’ 2 8 5
  • D 2 8 5
  • E โˆ’ 7 3

Q18:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅ+7๐‘ฅ6๐‘ฅ+25๐‘ฅ+4รท6๐‘ฅโˆ’๐‘ฅ36๐‘ฅโˆ’1๏Šจ๏Šจ๏Šจ๏Šจ.

  • A ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 7 ๐‘ฅ โˆ’ 4
  • B ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 7 ๐‘ฅ + 4
  • C ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 7 ๐‘ฅ ๐‘ฅ + 4 ๐‘ฅ ๏Šจ ๏Šจ
  • D ๐‘› ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 4 ๐‘ฅ + 7
  • E ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 4 ๐‘ฅ + 7

Q19:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅ+4๐‘ฅโˆ’12๐‘ฅโˆ’36รท5๐‘ฅโˆ’10๐‘ฅโˆ’12๐‘ฅ+36๏Šจ๏Šจ๏Šจ.

  • A ๐‘› ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 6 5
  • B ๐‘› ( ๐‘ฅ ) = 5 ๐‘ฅ โˆ’ 6
  • C ๐‘› ( ๐‘ฅ ) = ๐‘ฅ + 6 5
  • D ๐‘› ( ๐‘ฅ ) = 1 5 ( ๐‘ฅ โˆ’ 6 )
  • E ๐‘› ( ๐‘ฅ ) = 5 ๐‘ฅ + 6

Q20:

Simplify the function ๐‘›(๐‘ฅ)=9๐‘ฅ+72๐‘ฅ+1รท9๐‘ฅ+725๐‘ฅ+5.

  • A ๐‘› ( ๐‘ฅ ) = 5 8 1
  • B ๐‘› ( ๐‘ฅ ) = 8 1 5
  • C ๐‘› ( ๐‘ฅ ) = 5
  • D ๐‘› ( ๐‘ฅ ) = 1 5

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