Worksheet: Higher-Degree Polynomial Equations

In this worksheet, we will practice using different strategies for solving polynomial equations of degree greater than two.

Q1:

Solve the equation ( 3 𝑥 1 ) ( 5 𝑥 + 6 ) ( 3 𝑥 4 ) ( 8 𝑥 + 7 ) = 0 .

  • A 𝑥 = 1 , 𝑥 = 6 , 𝑥 = 4 , 𝑥 = 7
  • B 𝑥 = 1 3 , 𝑥 = 6 5 , 𝑥 = 4 3 , 𝑥 = 7 8
  • C 𝑥 = 1 , 𝑥 = 6 , 𝑥 = 4 , 𝑥 = 7
  • D 𝑥 = 1 3 , 𝑥 = 6 5 , 𝑥 = 4 3 , 𝑥 = 7 8
  • E 𝑥 = 3 , 𝑥 = 5 6 , 𝑥 = 3 4 , 𝑥 = 8 7

Q2:

Which of the following is a factorized form of 𝑥 + 2 𝑥 1 6 𝑥 3 2 3 2 ?

  • A ( 𝑥 4 ) ( 𝑥 + 2 ) 2
  • B ( 𝑥 + 4 ) ( 𝑥 2 ) 2
  • C ( 𝑥 + 4 ) ( 𝑥 4 ) ( 𝑥 2 )
  • D ( 𝑥 + 4 ) ( 𝑥 4 ) ( 𝑥 + 2 )
  • E ( 𝑥 + 4 ) ( 𝑥 + 2 )

Q3:

Given that 𝑓 ( 𝑥 ) = 𝑥 + 3 𝑥 1 3 𝑥 1 5 3 2 and 𝑓 ( 1 ) = 0 , find the other roots of 𝑓 ( 𝑥 ) .

  • A 𝑥 = 3 , 𝑥 = 5
  • B 𝑥 = 3 , 𝑥 = 5
  • C 𝑥 = 2 , 𝑥 = 6
  • D 𝑥 = 3 , 𝑥 = 5
  • E 𝑥 = 2 , 𝑥 = 6

Q4:

Which of the following is a factorized form of 3 𝑥 + 6 𝑥 5 7 𝑥 6 0 3 2 ?

  • A ( 3 𝑥 3 ) ( 𝑥 4 ) ( 𝑥 + 5 )
  • B ( 3 𝑥 3 ) ( 𝑥 + 4 ) ( 𝑥 5 )
  • C ( 3 𝑥 + 3 ) ( 𝑥 + 4 ) ( 𝑥 + 5 )
  • D ( 3 𝑥 + 3 ) ( 𝑥 4 ) ( 𝑥 + 5 )
  • E ( 3 𝑥 + 3 ) ( 𝑥 4 ) ( 𝑥 5 )

Q5:

By factoring, find all the solutions to 𝑥 + 2 𝑥 1 7 𝑥 1 8 𝑥 + 7 2 = 0 4 3 2 , given that ( 𝑥 3 ) and ( 𝑥 + 4 ) are factors of 𝑥 + 2 𝑥 1 7 𝑥 1 8 𝑥 + 7 2 4 3 2 .

  • A 𝑥 = 3 , 𝑥 = 4 , 𝑥 = 2 , 𝑥 = 3
  • B 𝑥 = 3 , 𝑥 = 4 , 𝑥 = 2 , 𝑥 = 3
  • C 𝑥 = 3 , 𝑥 = 4 , 𝑥 = 2 , 𝑥 = 3
  • D 𝑥 = 3 , 𝑥 = 4 , 𝑥 = 2 , 𝑥 = 3
  • E 𝑥 = 3 , 𝑥 = 4 , 𝑥 = 2

Q6:

Find the values of 𝑎 , 𝑏 , and 𝑐 given that ( 𝑥 + 3 ) , ( 𝑥 2 ) , and ( 𝑥 + 4 ) are factors of 𝑥 + 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 3 2 .

  • A 𝑎 = 5 , 𝑏 = 2 , 𝑐 = 2 4
  • B 𝑎 = 5 , 𝑏 = 2 , 𝑐 = 2 4
  • C 𝑎 = 5 , 𝑏 = 2 , 𝑐 = 2 4
  • D 𝑎 = 5 , 𝑏 = 2 , 𝑐 = 2 4
  • E 𝑎 = 5 , 𝑏 = 2 , 𝑐 = 2 4

Q7:

How many roots does the polynomial 3 𝑥 2 𝑥 + 𝑥 + 4 𝑥 2 6 3 2 have?

Q8:

Solve the equation ( 𝑥 1 ) ( 𝑥 + 6 ) ( 𝑥 4 ) ( 𝑥 + 7 ) = 0 .

  • A 𝑥 = 1 , 𝑥 = 6 , 𝑥 = 4 , 𝑥 = 7
  • B 𝑥 = 1 , 𝑥 = 6 , 𝑥 = 4 , 𝑥 = 7
  • C 𝑥 = 1 , 𝑥 = 6 , 𝑥 = 4 , 𝑥 = 7
  • D 𝑥 = 1 , 𝑥 = 6 , 𝑥 = 4 , 𝑥 = 7
  • E 𝑥 = 1 , 𝑥 = 6 , 𝑥 = 4 , 𝑥 = 7

Q9:

Factorise the expression 4 𝑐 2 8 6 𝑐 6 3 fully.

  • A 2 2 𝑐 + 7 𝑐 2 3 3
  • B 2 𝑐 7 2 𝑐 + 2 3 3
  • C 4 𝑐 1 4 𝑐 2 3 3
  • D 2 2 𝑐 7 𝑐 + 2 3 3
  • E 2 𝑐 + 7 2 𝑐 + 4 3 3

Q10:

Factorise the expression 7 5 𝑏 𝑚 + 6 0 𝑏 𝑚 + 1 2 𝑏 4 2 fully.

  • A 3 𝑏 ( 5 𝑚 2 ) 2
  • B 5 𝑏 5 𝑚 2 2 2
  • C 𝑏 ( 5 𝑚 + 2 ) 2
  • D 3 𝑏 5 𝑚 + 2 2 2
  • E 𝑏 5 𝑚 2 2 2

Q11:

Find the solution set of 𝑥 2 5 𝑥 + 1 4 4 = 0 in .

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

Q12:

Given that 𝑥 is in , find the value of 𝑥 which satisfies the following equation 3 𝑥 6 𝑥 + 9 = 0 . Give your answer to the nearest hundredth.

Q13:

Find the solution set of the equation 𝑥 5 0 6 𝑥 5 8 = 0 in .

  • A 5 8 , 5 8 , 5 0 6
  • B 5 8 , 5 0 6
  • C 5 8 , 5 0 6
  • D 5 8 , 5 8 , 5 0 6
  • E 5 0 6 , 5 0 6 , 5 8

Q14:

According to the fundamental theorem of algebra, how many complex solutions does the equation 𝑥 + 3 𝑥 3 𝑥 + 2 = 0 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 𝑥 + 𝑥 = 7 have?

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