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.

  • A๐‘ฅ(๐‘ฅ+3)3๐‘ฅโˆ’1๏Šจ
  • B๐‘ฅ(๐‘ฅ+3)๐‘ฅ+1๏Šจ
  • C๐‘ฅ(๐‘ฅ+3)3๐‘ฅโˆ’2๏Šจ
  • D๐‘ฅ(๐‘ฅ+3)4(3๐‘ฅโˆ’1)๏Šจ
  • E25(๐‘ฅโˆ’3)(๐‘ฅ+3)4๐‘ฅ(3๐‘ฅโˆ’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.

  • A3(2๐‘ฅโˆ’1)๐‘ฅ๏Šจ
  • B3(3๐‘ฅโˆ’1)๐‘ฅ๏Šจ
  • C๐‘ฅ+32๐‘ฅ๏Šจ
  • D3(๐‘ฅโˆ’1)๐‘ฅ๏Šจ
  • E2๐‘ฅ+32๐‘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.

  • A๐‘›(๐‘ฅ)=๐‘ฅโˆ’6(๐‘ฅ+4)(7๐‘ฅ+6), domain =โ„โˆ’๏ฌโˆ’4,โˆ’67๏ธ
  • B๐‘›(๐‘ฅ)=๐‘ฅโˆ’6(๐‘ฅโˆ’4)(7๐‘ฅโˆ’6), domain =โ„โˆ’๏ฌโˆ’9,โˆ’5,โˆ’4,โˆ’67๏ธ
  • C๐‘›(๐‘ฅ)=๐‘ฅโˆ’6(๐‘ฅ+4)(7๐‘ฅ+6), domain =โ„โˆ’๏ฌโˆ’9,โˆ’5,โˆ’4,โˆ’67๏ธ
  • D๐‘›(๐‘ฅ)=๐‘ฅ+6(๐‘ฅ+4)(7๐‘ฅ+6), domain =โ„โˆ’๏ฌโˆ’4,โˆ’67๏ธ
  • E๐‘›(๐‘ฅ)=๐‘ฅ+6(๐‘ฅ+4)(7๐‘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.

  • A๐‘›(๐‘ฅ)=๐‘ฅโˆ’49๐‘ฅ(๐‘ฅ+4)(2๐‘ฅ+9)
  • B๐‘›(๐‘ฅ)=(๐‘ฅโˆ’4)(2๐‘ฅ+9)9๐‘ฅ(๐‘ฅ+4)
  • C๐‘›(๐‘ฅ)=(๐‘ฅ+4)(2๐‘ฅโˆ’9)9๐‘ฅ(๐‘ฅโˆ’4)
  • D๐‘›(๐‘ฅ)=๐‘ฅ+49๐‘ฅ(๐‘ฅโˆ’4)(2๐‘ฅโˆ’9)
  • E๐‘›(๐‘ฅ)=(๐‘ฅ+4)(2๐‘ฅโˆ’9)๐‘ฅ(๐‘ฅโˆ’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.

  • Aโˆ’12
  • Bundefined
  • Cโˆ’188
  • Dโˆ’2

Q6:

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

  • A42๐‘ฅโˆ’105๐‘ฅ+42๐‘ฅ6๐‘ฅโˆ’7๐‘ฅ+2๏Šช๏Šฉ๏Šจ๏Šจ
  • B3๐‘ฅ(๐‘ฅโˆ’2)3๐‘ฅโˆ’4๏Šจ
  • C3๐‘ฅ(2๐‘ฅโˆ’1)7(3๐‘ฅโˆ’2)(๐‘ฅโˆ’2)๏Šจ๏Šจ
  • D7๐‘ฅ(๐‘ฅโˆ’2)3๐‘ฅโˆ’2๏Šจ
  • E21๐‘ฅ(๐‘ฅโˆ’2)3๐‘ฅโˆ’2๏Šจ

Q7:

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

  • A๐‘ฅ(4๐‘ฅโˆ’3)(2๐‘ฅโˆ’5)2(2๐‘ฅโˆ’1)
  • B๐‘ฅ(4๐‘ฅโˆ’5)(2๐‘ฅโˆ’3)2(2๐‘ฅโˆ’1)
  • C๐‘ฅ(4๐‘ฅโˆ’3)(2๐‘ฅโˆ’5)2(2๐‘ฅโˆ’1)๏Šจ
  • D8๐‘ฅโˆ’26๐‘ฅ+15๐‘ฅ8๐‘ฅโˆ’8๐‘ฅ+2๏Šฉ๏Šจ๏Šจ
  • E(4๐‘ฅโˆ’3)(2๐‘ฅโˆ’5)2(2๐‘ฅโˆ’1)๏Šจ

Q8:

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

  • A14๐‘ฅโˆ’3๐‘ฅ8๐‘ฅ+40๏Šจ
  • B7๐‘ฅโˆ’3๐‘ฅ8๐‘ฅโˆ’20๏Šจ
  • C7๐‘ฅ(2๐‘ฅโˆ’1)8(๐‘ฅ+5)
  • D7๐‘ฅ(2๐‘ฅโˆ’3)2(๐‘ฅโˆ’5)(2๐‘ฅโˆ’1)๏Šจ
  • E7๐‘ฅ(2๐‘ฅโˆ’1)8(๐‘ฅโˆ’5)

Q9:

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

  • Aโ„โˆ’{โˆ’6,โˆ’5}
  • Bโ„โˆ’{5,6}
  • Cโ„โˆ’{6}
  • Dโ„โˆ’{5}
  • Eโ„

Q10:

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

  • A1625๐‘ฅ๏Šจ
  • B64125๐‘ฅ๏Šฉ
  • C64125
  • D64125๐‘ฅ
  • E45๐‘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.

  • A๐‘›(๐‘ฅ)=โˆ’7๐‘ฅ, domain =โ„โˆ’{0}
  • B๐‘›(๐‘ฅ)=โˆ’7๐‘ฅ, domain =โ„โˆ’{โˆ’8,0,8}
  • C๐‘›(๐‘ฅ)=โˆ’17๐‘ฅ, domain =โ„โˆ’{โˆ’8,0,8}
  • D๐‘›(๐‘ฅ)=7๐‘ฅ, domain =โ„โˆ’{โˆ’8,0,8}
  • E๐‘›(๐‘ฅ)=7๐‘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.

  • A๐‘›(๐‘ฅ)=๐‘ฅ2(๐‘ฅ+3), domain =โ„โˆ’{โˆ’7,0}
  • B๐‘›(๐‘ฅ)=๐‘ฅ+32๐‘ฅ, domain =โ„โˆ’{0}
  • C๐‘›(๐‘ฅ)=๐‘ฅ+32๐‘ฅ, domain =โ„โˆ’{โˆ’7,0}
  • D๐‘›(๐‘ฅ)=2๐‘ฅ๐‘ฅ+3, domain =โ„โˆ’{0}
  • E๐‘›(๐‘ฅ)=2๐‘ฅ๐‘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.

  • A๐‘›(๐‘ฅ)=17, domain =โ„
  • B๐‘›(๐‘ฅ)=17, domain =โ„โˆ’{6}
  • C๐‘›(๐‘ฅ)=7, domain =โ„โˆ’{6}
  • D๐‘›(๐‘ฅ)=16, domain =โ„โˆ’{6}
  • E๐‘›(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.

  • Aโ„โˆ’{โˆ’3,โˆ’2}
  • Bโ„
  • Cโ„โˆ’{โˆ’2,2}
  • Dโ„โˆ’{โˆ’2,2,3}
  • Eโ„โˆ’{โˆ’3,โˆ’2,2}

Q15:

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

  • A32
  • B18
  • Cโˆ’16
  • Dโˆ’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).

  • Aโ„โˆ’{โˆ’6}
  • Bโ„โˆ’{3,6}
  • Cโ„โˆ’{0,6}
  • Dโ„โˆ’{6}
  • Eโ„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.

  • Aโˆ’283
  • B283
  • Cโˆ’285
  • D285
  • Eโˆ’73

Q18:

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

  • A๐‘›(๐‘ฅ)=๐‘ฅ+7๐‘ฅโˆ’4
  • B๐‘›(๐‘ฅ)=๐‘ฅ+7๐‘ฅ+4
  • C๐‘›(๐‘ฅ)=๐‘ฅ+7๐‘ฅ๐‘ฅ+4๐‘ฅ๏Šจ๏Šจ
  • D๐‘›(๐‘ฅ)=๐‘ฅโˆ’4๐‘ฅ+7
  • E๐‘›(๐‘ฅ)=๐‘ฅ+4๐‘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.

  • A๐‘›(๐‘ฅ)=๐‘ฅโˆ’65
  • B๐‘›(๐‘ฅ)=5๐‘ฅโˆ’6
  • C๐‘›(๐‘ฅ)=๐‘ฅ+65
  • D๐‘›(๐‘ฅ)=15(๐‘ฅโˆ’6)
  • E๐‘›(๐‘ฅ)=5๐‘ฅ+6

Q20:

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

  • A๐‘›(๐‘ฅ)=581
  • B๐‘›(๐‘ฅ)=815
  • C๐‘›(๐‘ฅ)=5
  • D๐‘›(๐‘ฅ)=15

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