Worksheet: Zeros of Polynomial Functions

In this worksheet, we will practice finding the set of zeros of a quadratic, cubic, or higher-degree polynomial function.

Q1:

Find, by factoring, the zeros of the function 𝑓(𝑥)=𝑥+2𝑥35.

  • A 7 , 5
  • B 7 , 5
  • C 5 , 7
  • D 6 , 8
  • E 5 , 7

Q2:

What are the zeros of the function 𝑓(𝑥)=2(𝑥1)7?

  • A 1 + 7 2 and 172
  • B 1 + 7 2 and 172
  • C 1 7 2 and 172
  • D 1 + 7 2 and 172
  • E 1 + 7 2 and 172

Q3:

Find, by factoring, the zeros of the function 𝑓(𝑦)=𝑦+8𝑦+7.

  • A 7 , 1
  • B 8 , 1
  • C 7 , 1
  • D 7 , 1
  • E 1 , 8

Q4:

Find the set of zeros of the function 𝑓(𝑥)=𝑥17𝑥+16.

  • A { 4 , 1 }
  • B { 1 , 4 }
  • C { 1 }
  • D { 4 }
  • E { 4 , 1 , 1 , 4 }

Q5:

If the set of zeros of the function 𝑓(𝑥)=𝑥+𝑏𝑥+343 is {8,8}, find the value of 𝑏.

Q6:

Find, by factoring, the zeros of the function 𝑓(𝑥)=9𝑥+9𝑥40.

  • A 5 , 8
  • B 5 3 , 8 3
  • C 5 3 , 8 3
  • D 5 , 8
  • E 5 3 , 8 3

Q7:

Find the set of zeros of the function 𝑓(𝑥)=𝑥𝑥812𝑥81.

  • A { 2 , 9 }
  • B { 9 , 9 }
  • C { 9 , 2 , 9 }
  • D { 9 , 2 , 9 }
  • E { 2 , 9 }

Q8:

Find the set of zeros of the function 𝑓(𝑥)=13(𝑥4).

  • A { 4 }
  • B 1 3 , 4
  • C { 4 }
  • D 1 3 , 4

Q9:

Find the set of zeros of the function 𝑓(𝑥)=𝑥1𝑥4.

  • A { 2 , 2 }
  • B { 1 }
  • C { 2 , 2 }
  • D { 2 , 1 , 2 }
  • E { 1 }

Q10:

𝑓 ( 𝑥 ) = 4 𝑥 + 𝑏 𝑥 5 𝑥 + 4 2 , 𝑓 ( 4 ) = 2 2 , and 𝑓(2)=0. Find the other roots of 𝑓(𝑥) and the value of 𝑏.

  • A 𝑏 = 1 6 , 𝑥 = 3 2 , 𝑥 = 7 2
  • B 𝑏 = 1 6 , 𝑥 = 3 2 , 𝑥 = 7 2
  • C 𝑏 = 1 6 , 𝑥 = 3 2 , 𝑥 = 7 2
  • D 𝑏 = 1 6 , 𝑥 = 3 2
  • E 𝑏 = 1 6 , 𝑥 = 3 2 , 𝑥 = 7 2

Q11:

Find all the zeros of 𝑓(𝑥)=𝑥+5𝑥9𝑥45 and state their multiplicities.

  • A 𝑥 = 3 with multiplicity 1, 𝑥=5 with multiplicity 1, 𝑥=3 with multiplicity 1
  • B 𝑥 = 3 with multiplicity 2, 𝑥=3 with multiplicity 2
  • C 𝑥 = 3 with multiplicity 1, 𝑥=5 with multiplicity 1
  • D 𝑥 = 5 with multiplicity 1, 𝑥=3 with multiplicity 2

Q12:

Which of the following functions have the same set of zeros?

  • A 𝑘 ( 𝑥 ) = 𝑥 + 1 0 𝑥 and 𝑓(𝑥)=𝑥20𝑥+100𝑥
  • B 𝑘 ( 𝑥 ) = 𝑥 1 0 𝑥 and 𝑓(𝑥)=𝑥+20𝑥+100𝑥
  • C 𝑘 ( 𝑥 ) = 𝑥 1 0 𝑥 and 𝑓(𝑥)=𝑥+20𝑥+100𝑥
  • D 𝑘 ( 𝑥 ) = 𝑥 + 1 0 𝑥 and 𝑓(𝑥)=𝑥+20𝑥+100𝑥
  • E 𝑘 ( 𝑥 ) = 𝑥 + 1 0 𝑥 and 𝑓(𝑥)=𝑥20𝑥+100𝑥

Q13:

Find the set of zeros of the function 𝑓(𝑥)=6𝑥𝑥+64.

  • A
  • B { 0 }
  • C { 8 }
  • D { 0 , 8 , 8 }
  • E { 8 , 8 }

Q14:

The function 𝑓(𝑥)=𝑎𝑥+54𝑥+81 and the function 𝑔(𝑥)=𝑎𝑥+9 have the same set of zeros. Find 𝑎 and the set of zeros.

  • A 𝑎 = 9 , 𝑧 ( 𝑓 ) = { 3 }
  • B 𝑎 = 9 , 𝑧 ( 𝑓 ) = { 3 }
  • C 𝑎 = 3 , 𝑧 ( 𝑓 ) = 1 3
  • D 𝑎 = 3 , 𝑧 ( 𝑓 ) = { 3 }
  • E 𝑎 = 3 , 𝑧 ( 𝑓 ) = { 3 }

Q15:

Find the set of zeros of the function 𝑓(𝑥)=9𝑥+225𝑥.

  • A { 5 , 5 , 9 }
  • B { 0 , 5 }
  • C { 5 , 0 , 5 }
  • D { 9 , 5 , 5 }
  • E { 5 , 5 }

Q16:

Find the set of zeros of the function 𝑓(𝑥)=7𝑥112𝑥.

  • A { 0 , 4 , 4 }
  • B { 7 , 4 , 4 }
  • C { 7 , 4 , 4 }
  • D { 4 , 4 }
  • E { 0 , 4 }

Q17:

Given the function 𝑓(𝑥)=𝑥5𝑥+2𝑥+8, determine the vertical and horizontal intercepts.

  • AVertical intercept: 𝑥=1 or 2; horizontal intercept: 𝑦=8
  • BVertical intercept: 𝑥=1,2, or 4; horizontal intercept: 𝑦=8
  • CVertical intercept: 𝑦=0; horizontal intercept: 𝑥=1 or 2
  • DVertical intercept: 𝑦=8; horizontal intercept: 𝑥=1,2, or 4

Q18:

What is the set of zeros of the function 𝑛(𝑥)=𝑥𝑥+76𝑥+7?

  • A { 6 }
  • B { 7 }
  • C { 7 }
  • D { 6 }
  • E { 7 }

Q19:

Find the set of zeros of the function 𝑓(𝑥)=𝑥24.

  • A { 2 4 }
  • B { 2 4 }
  • C
  • D { 0 , 2 4 }
  • E { 0 , 2 4 }

Q20:

Consider the function 𝑘(𝑥)=5𝑥+2𝑥30𝑥88𝑥+40.

Given that one zero of 𝑘(𝑥) is 13𝑖, find all zeros of 𝑘(𝑥) using synthetic division.

  • A 1 3 𝑖 , 1 + 3 𝑖 , 1 , 4 5
  • B 1 3 𝑖 , 1 + 3 𝑖 , 2 , 2 5
  • C 1 3 𝑖 , 1 + 3 𝑖 , 4 , 1 5
  • D 1 3 𝑖 , 1 + 3 𝑖 , 2 , 2 5
  • E 1 3 𝑖 , 1 + 3 𝑖 , 4 , 1 5

Write the linear factorization of 𝑘(𝑥).

  • A 𝑘 ( 𝑥 ) = ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 ) ( 5 𝑥 + 4 ) ( 𝑥 1 )
  • B 𝑘 ( 𝑥 ) = ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 ) ( 5 𝑥 + 1 ) ( 𝑥 4 )
  • C 𝑘 ( 𝑥 ) = ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 ) ( 5 𝑥 1 ) ( 𝑥 + 4 )
  • D 𝑘 ( 𝑥 ) = ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 ) ( 5 𝑥 2 ) ( 𝑥 + 2 )
  • E 𝑘 ( 𝑥 ) = ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 ) ( 5 𝑥 + 2 ) ( 𝑥 2 )

Q21:

Consider (𝑥)=16𝑥88𝑥+313𝑥348𝑥+117.

Given that one zero of multiplicity 2 of (𝑥) is 34, find all zeros of (𝑥) using synthetic division.

  • A 3 4 , 2 3 𝑖 , 2 + 3 𝑖
  • B 3 4 , 2 3 𝑖 , 2 + 3 𝑖
  • C 3 4 , 2 6 𝑖 , 2 + 6 𝑖
  • D 3 4 , 2 6 𝑖 , 2 + 6 𝑖

Write the linear factorization of (𝑥).

  • A ( 𝑥 ) = ( 4 𝑥 3 ) ( 𝑥 2 + 3 𝑖 ) ( 𝑥 2 3 𝑖 )
  • B ( 𝑥 ) = ( 4 𝑥 3 ) ( 𝑥 + 2 + 3 𝑖 ) ( 𝑥 + 2 3 𝑖 )
  • C ( 𝑥 ) = ( 4 𝑥 3 ) ( 𝑥 + 2 + 6 𝑖 ) ( 𝑥 + 2 6 𝑖 )
  • D ( 𝑥 ) = ( 4 𝑥 3 ) ( 𝑥 2 + 6 𝑖 ) ( 𝑥 2 6 𝑖 )

Q22:

Consider the function 𝑓(𝑥)=𝑥6𝑥+14𝑥32𝑥40.

Given that one zero of 𝑓(𝑥) is 222, find all zeros of 𝑓(𝑥) using synthetic division.

  • A 2 2 2 , 2 + 2 2 , 1 3 𝑖 , 1 3 𝑖
  • B 2 2 2 , 2 2 2 , 1 3 𝑖 , 1 3 𝑖
  • C 2 2 2 , 2 2 2 , 1 3 𝑖 , 1 + 3 𝑖
  • D 2 2 2 , 2 + 2 2 , 1 3 𝑖 , 1 + 3 𝑖
  • E 2 2 2 , 2 + 2 2 , 1 + 3 𝑖 , 1 + 3 𝑖

Write the linear factorization of 𝑓(𝑥).

  • A 𝑓 ( 𝑥 ) = 𝑥 2 + 2 2 𝑥 2 2 2 ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 )
  • B 𝑓 ( 𝑥 ) = 𝑥 2 + 2 2 𝑥 2 2 2 ( 𝑥 + 1 + 3 𝑖 ) ( 𝑥 1 + 3 𝑖 )
  • C 𝑓 ( 𝑥 ) = 𝑥 2 + 2 2 𝑥 2 2 2 ( 𝑥 + 1 3 𝑖 ) ( 𝑥 1 3 𝑖 )
  • D 𝑓 ( 𝑥 ) = 𝑥 + 2 + 2 2 𝑥 2 + 2 2 ( 𝑥 1 + 3 𝑖 ) ( 𝑥 + 1 + 3 𝑖 )
  • E 𝑓 ( 𝑥 ) = 𝑥 2 + 2 2 𝑥 + 2 + 2 2 ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 )

Q23:

Consider 𝑔(𝑥)=𝑥+6𝑥+38𝑥+24𝑥+136.

Given that one zero of 𝑔(𝑥) is 3+5𝑖, find all zeros of 𝑔(𝑥) using synthetic division.

  • A 3 + 5 𝑖 , 3 5 𝑖 , 2 , 2
  • B 3 + 5 𝑖 , 3 5 𝑖 , 2
  • C 3 + 5 𝑖 , 3 5 𝑖 , 2 𝑖 , 2 𝑖
  • D 3 + 5 𝑖 , 3 + 5 𝑖 , 2 𝑖 , 2 𝑖

Write the linear factorization of 𝑔(𝑥).

  • A 𝑔 ( 𝑥 ) = ( 𝑥 + 3 5 𝑖 ) ( 𝑥 + 3 + 5 𝑖 ) ( 𝑥 + 2 ) ( 𝑥 2 )
  • B 𝑔 ( 𝑥 ) = ( 𝑥 3 5 𝑖 ) ( 𝑥 + 3 5 𝑖 ) ( 𝑥 + 2 𝑖 ) ( 𝑥 2 𝑖 )
  • C 𝑔 ( 𝑥 ) = ( 𝑥 + 3 5 𝑖 ) ( 𝑥 + 3 + 5 𝑖 ) ( 𝑥 + 2 𝑖 ) ( 𝑥 2 𝑖 )
  • D 𝑔 ( 𝑥 ) = ( 𝑥 + 3 5 𝑖 ) ( 𝑥 + 3 + 5 𝑖 ) ( 𝑥 2 )

Q24:

If 𝐹 is an integral domain and 𝑓(𝑥) is a polynomial with coefficients in 𝐹, then which of the following is true of 𝑓(𝑥)?

  • A 𝑓 ( 𝑥 ) is a constant polynomial.
  • B 𝑓 ( 𝑥 ) is irreducible.
  • C 𝑓 ( 𝑥 ) must have a unique factorization.
  • D 𝑓 ( 𝑥 ) is either irreducible or factorable.

Q25:

Given that 𝑓(𝑥)=𝑥+3𝑥13𝑥15 and 𝑓(1)=0, find the other roots of 𝑓(𝑥).

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

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