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Worksheet: Higher-Degree Polynomial Equations

Q1:

Solve the equation ( 3 ๐‘ฅ โˆ’ 1 ) ( 5 ๐‘ฅ + 6 ) ( 3 ๐‘ฅ โˆ’ 4 ) ( 8 ๐‘ฅ + 7 ) = 0 .

  • A ๐‘ฅ = 1 , ๐‘ฅ = โˆ’ 6 , ๐‘ฅ = 4 , ๐‘ฅ = โˆ’ 7
  • B ๐‘ฅ = โˆ’ 1 3 , ๐‘ฅ = 6 5 , ๐‘ฅ = โˆ’ 4 3 , ๐‘ฅ = 7 8
  • C ๐‘ฅ = โˆ’ 1 , ๐‘ฅ = 6 , ๐‘ฅ = โˆ’ 4 , ๐‘ฅ = 7
  • D ๐‘ฅ = 1 3 , ๐‘ฅ = โˆ’ 6 5 , ๐‘ฅ = 4 3 , ๐‘ฅ = โˆ’ 7 8
  • E ๐‘ฅ = 3 , ๐‘ฅ = โˆ’ 5 6 , ๐‘ฅ = 3 4 , ๐‘ฅ = โˆ’ 8 7

Q2:

Which of the following is a factorized form of ๐‘ฅ + 2 ๐‘ฅ โˆ’ 1 6 ๐‘ฅ โˆ’ 3 2 3 2 ?

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

Q3:

Which of the following is a factorized form of 3 ๐‘ฅ + 6 ๐‘ฅ โˆ’ 5 7 ๐‘ฅ โˆ’ 6 0 3 2 ?

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

Q4:

Factor the expression 7 5 ๐‘ ๐‘š + 6 0 ๐‘ ๐‘š + 1 2 ๐‘ 4 2 fully.

  • A 3 ๐‘ ( 5 ๐‘š โˆ’ 2 ) 2
  • B 5 ๐‘ ๏€น 5 ๐‘š โˆ’ 2 ๏… 2 2
  • C ๐‘ ( 5 ๐‘š + 2 ) 2
  • D 3 ๐‘ ๏€น 5 ๐‘š + 2 ๏… 2 2
  • E ๐‘ ๏€น 5 ๐‘š โˆ’ 2 ๏… 2 2

Q5:

Find the solution set of ๐‘ฅ โˆ’ 2 5 ๐‘ฅ + 1 4 4 = 0 4 2 in โ„ .

  • A { 8 , โˆ’ 8 , 1 8 , โˆ’ 1 8 }
  • B { โˆ’ 1 6 , โˆ’ 9 }
  • C { โˆ’ 8 , โˆ’ 1 8 }
  • D { 4 , โˆ’ 4 , 3 , โˆ’ 3 }
  • E { 4 , 3 }

Q6:

Factor the expression 4 ๐‘ โˆ’ 2 8 โˆ’ 6 ๐‘ 6 3 fully.

  • A 2 ๏€น 2 ๐‘ + 7 ๏… ๏€น ๐‘ โˆ’ 2 ๏… 3 3
  • B 2 ๏€น ๐‘ โˆ’ 7 ๏… ๏€น 2 ๐‘ + 2 ๏… 3 3
  • C ๏€น 4 ๐‘ โˆ’ 1 4 ๏… ๏€น ๐‘ โˆ’ 2 ๏… 3 3
  • D 2 ๏€น 2 ๐‘ โˆ’ 7 ๏… ๏€น ๐‘ + 2 ๏… 3 3
  • E ๏€น 2 ๐‘ + 7 ๏… ๏€น 2 ๐‘ + 4 ๏… 3 3

Q7:

Given that ๐‘ฅ is in โ„ , find the value of ๐‘ฅ which satisfies the following equation ๏€น 3 ๐‘ฅ โˆ’ 6 ๏… ๏€น ๐‘ฅ + 9 ๏… = 0 3 2 . Give your answer to the nearest hundredth.

Q8:

How many roots does the polynomial 3 ๐‘ฅ โˆ’ 2 ๐‘ฅ + ๐‘ฅ + 4 ๐‘ฅ โˆ’ 2 6 3 2 have?

Q9:

Find the solution set of the equation ๏€น ๐‘ฅ โˆ’ 5 0 6 ๏… ๏€น ๐‘ฅ โˆ’ 5 8 ๏… = 0 3 2 in โ„ .

  • A ๏ซ โˆš 5 8 , โˆ’ โˆš 5 8 , โˆ’ โˆš 5 0 6 ๏ท 3
  • B ๏ซ โˆš 5 8 , โˆš 5 0 6 ๏ท 3
  • C ๏ซ โˆ’ โˆš 5 8 , โˆ’ โˆš 5 0 6 ๏ท 3
  • D ๏ซ โˆš 5 8 , โˆ’ โˆš 5 8 , โˆš 5 0 6 ๏ท 3
  • E ๏ซ โˆš 5 0 6 , โˆ’ โˆš 5 0 6 , โˆš 5 8 ๏ท 3

Q10:

Solve the equation ( ๐‘ฅ โˆ’ 1 ) ( ๐‘ฅ + 6 ) ( ๐‘ฅ โˆ’ 4 ) ( ๐‘ฅ + 7 ) = 0 .

  • A ๐‘ฅ = โˆ’ 1 , ๐‘ฅ = โˆ’ 6 , ๐‘ฅ = โˆ’ 4 , ๐‘ฅ = โˆ’ 7
  • B ๐‘ฅ = โˆ’ 1 , ๐‘ฅ = 6 , ๐‘ฅ = โˆ’ 4 , ๐‘ฅ = 7
  • C ๐‘ฅ = 1 , ๐‘ฅ = 6 , ๐‘ฅ = 4 , ๐‘ฅ = 7
  • D ๐‘ฅ = 1 , ๐‘ฅ = โˆ’ 6 , ๐‘ฅ = 4 , ๐‘ฅ = โˆ’ 7
  • E ๐‘ฅ = 1 , ๐‘ฅ = โˆ’ 6 , ๐‘ฅ = โˆ’ 4 , ๐‘ฅ = โˆ’ 7

Q11:

Given that ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 3 ๐‘ฅ โˆ’ 1 3 ๐‘ฅ โˆ’ 1 5 3 2 and ๐‘“ ( โˆ’ 1 ) = 0 , find the other roots of ๐‘“ ( ๐‘ฅ ) .

  • A ๐‘ฅ = โˆ’ 3 , ๐‘ฅ = โˆ’ 5
  • B ๐‘ฅ = โˆ’ 3 , ๐‘ฅ = 5
  • C ๐‘ฅ = โˆ’ 2 , ๐‘ฅ = โˆ’ 6
  • D ๐‘ฅ = 3 , ๐‘ฅ = โˆ’ 5
  • E ๐‘ฅ = 2 , ๐‘ฅ = 6

Q12:

By factoring, find all the solutions to ๐‘ฅ + 2 ๐‘ฅ โˆ’ 1 7 ๐‘ฅ โˆ’ 1 8 ๐‘ฅ + 7 2 = 0 4 3 2 , given that ( ๐‘ฅ โˆ’ 3 ) and ( ๐‘ฅ + 4 ) are factors of ๐‘ฅ + 2 ๐‘ฅ โˆ’ 1 7 ๐‘ฅ โˆ’ 1 8 ๐‘ฅ + 7 2 4 3 2 .

  • A ๐‘ฅ = โˆ’ 3 , ๐‘ฅ = โˆ’ 4 , ๐‘ฅ = โˆ’ 2 , ๐‘ฅ = 3
  • B ๐‘ฅ = โˆ’ 3 , ๐‘ฅ = 4 , ๐‘ฅ = โˆ’ 2 , ๐‘ฅ = 3
  • C ๐‘ฅ = 3 , ๐‘ฅ = 4 , ๐‘ฅ = 2 , ๐‘ฅ = โˆ’ 3
  • D ๐‘ฅ = 3 , ๐‘ฅ = โˆ’ 4 , ๐‘ฅ = 2 , ๐‘ฅ = โˆ’ 3
  • E ๐‘ฅ = 3 , ๐‘ฅ = โˆ’ 4 , ๐‘ฅ = โˆ’ 2

Q13:

Find the values of ๐‘Ž , ๐‘ , and ๐‘ given that ( ๐‘ฅ + 3 ) , ( ๐‘ฅ โˆ’ 2 ) , and ( ๐‘ฅ + 4 ) are factors of ๐‘ฅ + ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ฅ + ๐‘ 3 2 .

  • A ๐‘Ž = 5 , ๐‘ = 2 , ๐‘ = โˆ’ 2 4
  • B ๐‘Ž = โˆ’ 5 , ๐‘ = โˆ’ 2 , ๐‘ = โˆ’ 2 4
  • C ๐‘Ž = 5 , ๐‘ = โˆ’ 2 , ๐‘ = 2 4
  • D ๐‘Ž = 5 , ๐‘ = โˆ’ 2 , ๐‘ = โˆ’ 2 4
  • E ๐‘Ž = โˆ’ 5 , ๐‘ = 2 , ๐‘ = 2 4

Q14:

According to the fundamental theorem of algebra, how many complex solutions does the equation ๐‘ฅ + 3 ๐‘ฅ โˆ’ 3 ๐‘ฅ + 2 = 0 ๏Šฎ ๏Šช ๏Šฉ have, counted with multiplicity?

  • AExactly two
  • BAt least one, but the exact number is unknown.
  • CInfinitely many
  • DEight

Q15:

How many roots does the equation ๐‘ฅ + ๐‘ฅ = 7 2 have?